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MODELOS DE ISING E POTTS ACOPLADOS ASTRIANGULAÇÕES DE LORENTZ

José Javier Cerda Hernández

Dissertação/Tese apresentadaao

Instituto de Matemática e Estatísticada

Universidade de São Paulopara

obtenção do títulode

Doutor em Ciências

Programa: Estatística

Orientador: Prof. Dr. Anatoli Iambartsev

Coorientador: Prof. Dr. Yuri Suhov

Durante o desenvolvimento deste trabalho o autor recebeu auxílio nanceiro da

CAPES/FAPESP

São Paulo, junho de 2014

MODELOS DE ISING E POTTS ACOPLADOS ASTRIANGULAÇÕES DE LORENTZ

Esta é a versão original da dissertação/tese elaborada pelo

candidato José Javier Cerda Hernández, tal como

submetida à Comissão Julgadora.

Agradecimentos

First of all I would like to thank my supervisors Anatoli Iambartsev and Yuri Suhov for

guiding me through this research and their professional advisory and patience, as well as for

giving me the freedom to follow dierent themes during my research........

This work was supported by CAPES and FAPESP (projects 2012/04372-7 and 2013/06179-

2). Further, the author thanks the IME at the University of São Paulo for warm hospitality.

..........

i

ii

Resumo

José Javier Cerda Hernández. MODELOS DE ISING E POTTS ACOPLADOS

AS TRIANGULAÇÕES DE LORENTZ. 2010. 91 f. Tese (Doutorado) - Instituto de

Matemática e Estatística, Universidade de São Paulo, São Paulo, 2010.

O objetivo principal da presente tese é pesquisar : Quais são as propriedades do modelo de

Ising e Potts acoplado ao emsemble de CDT? Para estudar o modelo usamos dois metodos:

(1) Matriz de transferencia e Theorema de Krein-Rutman. (2) Representação FK para o

modelo de Potts sobre CDT e dual de CDT.

Matriz de transferencia permite obter propriedades espectrais da Matriz de transferencia

utilisando o Teorema de Krein-Rutman [KR48] sobre operadores que conservam o cone

de funções positivas. Também obtemos propriedades asintóticas da função de partição e

das medidas de Gibbs. Esses propriedades permitem obter uma região onde a energia livre

converge. O segundo método permite obter uma região onde a curva crítica do modelo

pode estar localizada. Alem disso, também obtemos uma limitante superior e inferior para

a energia livre a volume innito.

Finalmente, utilizando argumentos de dualidade em grafos e expansão em alta temper-

atura estudamos o modelo de Potts acoplado com triangulações causais. Essa abordagem

permite generalizar o modelo, melhorar os resultados obtidos para o modelo de Ising e obter

novas limitantes, superior e inferior, para a energia livre e para a curva crítica. Alem do

mais, obtemos uma aproximação do autovalor maximal do operador de transferencia a baixa

temperatura.

Palavras-chave: dinâmica de triangulações causais, modelo de Ising, modelo de Potts,

medida de Gibbs, Teorema de Krein-Rutman, representação FK, modelo de Ising quântico.

iii

iv

Abstract

José Javier Cerda Hernández. Ising and Potts model coupled to Lorentzian triangu-

lations. 2014. 91 f. Tese (Doutorado) - Instituto de Matemática e Estatística, Universidade

de São Paulo, São Paulo, 2014.

The main objective of the present thesis is to investigate: What are the properties of

the Ising and Potts model coupled to a CDT emsemble? For that objetive, we used two

methods: (1) transfer matrix formalism and Krein-Rutman theory. (2) FK representation of

the q-state Potts model on CDTs and dual CDTs.

Transfer matrix formalism permite us obtain spectral properties of the transfer matrix

using the Krein-Rutman theorem [KR48] on operators preserving the cone of positive func-

tions. This yields results on convergence and asymptotic properties of the partition function

and the Gibbs measure and allows us to determine regions in the parameter quarter-plane

where the free energy converges. Second methods permite us determining a region in the

quadrant of parameters β, µ > 0 where the critical curve for the classical model can be

located. We also provide lower and upper bounds for the innite-volume free energy.

FInally, using arguments of duality on graph theory and hight-T expansion we study

the Potts model coupled to CDTs. This approach permite us improve the results obtained

for Ising model and obtain lower and upper bounds for the critical curve and free energy.

Moreover, we obtain an approximation of the maximal eigenvalue of the transfer matrix at

lower temperature.

Keywords: causal dynamical triangulation, Ising model, Potts model, Gibbs measure,

Krein-Rutman theory, FK representation, quantum Ising model.

v

vi

Contents

List of Figures ix

1 Introduction 1

1.1 Introduction and statement results . . . . . . . . . . . . . . . . . . . . . . . 1

2 Two-dimensional causal dynamical Triangulations 5

2.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Transfer matrix formalism for pure CDTs . . . . . . . . . . . . . . . . . . . . 7

3 Transfer matrix formalism for Ising model coupled to two-dimensional

CDT 13

3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 The transfer-matrix K and its powers KN . . . . . . . . . . . . . . . . . . . 17

3.3 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 FK representation for the Ising model coupled to CDT 27

4.1 The quantum Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 FK representation for Ising model coupled to CDT . . . . . . . . . . . . . . 29

4.3 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Proof of Theorem 4.3.1 and 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.1 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.2 Proof of Theorem 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Potts model coupled to CDTs and FK representation 41

5.1 Introduction and main results of this chapter . . . . . . . . . . . . . . . . . . 41

5.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2.1 A Potts model coupled to CDTs . . . . . . . . . . . . . . . . . . . . . 45

5.2.2 The FK-Potts model on Lorentzian triangulations . . . . . . . . . . . 46

5.2.3 The relation between the Potts model and FK-Potts model: Edwards-

Sokal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.4 Duality for FK-Potts model coupled to CDTs with periodic boundary

conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 The proof of Theorem 5.1.1 and rst bounds for the critical curve . . . . . . 52

vii

viii CONTENTS

5.4 High-T expansion of the Potts model and Proof of Theorem 5.1.2 . . . . . . 58

5.5 Connection between transfer matrix and FK representation . . . . . . . . . . 62

5.5.1 q = 2 (Ising) systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.5.2 q-Potts systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A The von Neumann-Schatten Classes of Operators 71

A.1 The space Cp and rst properties . . . . . . . . . . . . . . . . . . . . . . . . 71

A.2 The trace class C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.3 The Banach space Cp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.4 The Hilbert-Schmidt class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B Krein-Rutman theorem 75

B.1 Krein-Rutman Theorem and the Principal Eigenvalue . . . . . . . . . . . . . 75

Bibliography 77

List of Figures

1.1 A strip of a causal triangulation of S × [j, j + 1]. . . . . . . . . . . . . . . . 2

2.1 (a) A strip of a causal triangulation of S × [j, j + 1]. (b) Geometric represen-

tation of a CDT with periodic spatial boundary condition. . . . . . . . . . . 7

2.2 Tree parametrization of a causal dynamical triangulation. . . . . . . . . . . . 11

3.1 Illustration of the calculates (3.25) and (3.27). . . . . . . . . . . . . . . . . . 22

3.2 λQ = λ and λT are the maximal eigenvalues of the matrix Q and a related

matrix T respectively. The area above the black curve is where the condition

(3.20) holds true. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 A trajectory sample associated with a realization ξ = sii=1,...,n. Each tra-

jectory ϕ ∈ ψξ can be continuous or not at each arrival time s. In this case,

at arrival time sk−1 the trajectory ϕ do not have jump, and at arrival time

sk the trajectory ϕ have a jump. . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 In this gure, we show the Cluster Ct of a triangle t, and a graphic represen-

tation of relation t↔ t′, where↔ on right side in the gure, represent arrival

times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 The area above the minimum of the dotted curve I (graph of the function ψ

dened in (4.21)) and dash-dotted line II is where the limiting Gibbs proba-

bility measure exists and is unique. The critical curve lies in the region below

the dotted curve I and dash-dotted line II but above the continuous curve III

and dashed line IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1 Illustrating the region where the critical curve for Potts model coupled CDTs

and dual CDTs can be located. . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Geometric representation of a dual Lorentzian triangulation t∗ with periodic

spatial boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 (a) Geometric representation of a net (b) Geometric representation of a cycle

(c) None of cluster of w is a net or a cycle . . . . . . . . . . . . . . . . . . . 49

5.4 Examples of three subgraphs of A with 8 edges. It is clear that the term

ξ(e1, . . . , e8) depends of the topology of the subgraphs. . . . . . . . . . . . . 59

ix

x LIST OF FIGURES

5.5 Region where the critical curve of the Ising model coupled to dual CDTs can

be located. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6 The blue line is the simulation of ||A||2 = 1 for q = 4. Black line: µ∗ = 3 ln 2.

Green line: µ∗ = 32

ln(eβ∗ − 1

)+ ln 2. Red line: µ∗ = 3

2ln(42/3 + eβ

∗ − 1)

+ ln 2. 68

Chapter 1

Introduction

1.1 Introduction and statement results

Models of planar random geometry appear in physics in the context of two-dimensional

quantum gravity and provide an interplay between mathematical physics and probability

theory.

Causal dynamical triangulation (CDT), introduced by Ambjørn and Loll (see [AL98]),

together with its predecessor a dynamical triangulation (DT), constitute attemps to provide

a meaning to formal expressions appearing in the path integral quantisation of gravity (see

[ADJ97], [AJ06] for an overview). A causal triangulation is formed by triangulations of spa-

tial strips as illustrated in Figure 5.2. Note that the left and right boundaries of the spatial

strip are periodically identied. The idea is to regularise the path integral by approximating

the geometries emerging in the integration by CDTs. As a result, the path integral over

geometries is replaced with a sum over all possible triangulations where each conguration

is weighted by a Boltzmann factor e−µ|T |, with |T | standing for the size of the triangula-

tion and µ being the cosmological constant. The evaluation of the partition function was

reduced to a purely combinatorial problem that can be solved with the help of the early

work of Tutte [Tut62, Tut63]; alternatively, more powerful techniques were proposed, based

on random matrix models (see, e.g., [FGZJ95]) and bijections to well-labelled trees (see

[Sch97, BDG02]).

From a probabilistic point of view there has recently been an increasing interest in DT,

most notably through the work of Angel and Schramm on a uniform measure on innite

planar triangulations [AS03], as well as through the work of Le Gall, Miermont and collab-

orators on Brownian maps (see [GG11] for a recent review).

From a physical point of view it is interesting to study various models of matter, such

as the Ising and Potts model, coupled to the CDT. An interesting question is: What are

the properties of the Ising and Potts model coupled to a CDT ensemble? It is still random

and allows for a back-reaction of the spin system with the quantum geometry. Monte Carlo

simulations [AAL99] (see also [BL07, AALP08]) give a strong evidence that critical exponents

1

2 INTRODUCTION 1.1

root"up"

down"

S ×[ j, j +1]

Figure 1.1: A strip of a causal triangulation of S × [j, j + 1].

of the Ising model coupled to CDT are identical to the Onsager values. The calculation of the

partition function in this case also reduces to a combinatorial problem. It was rst solved in

[Kaz86, BK87] by using random matrix models and later by using a bijection to well-labelled

trees [BMS11]. It is interesting that the solution here is much simpler than in the case of

a at triangular or square lattice as given by Onsager [Ons44]. For the 2-state Potts model

(Ising model) coupled to CDTS some progress has been recently made on existence of Gibbs

measures and phase transitions (see [AAL99], [BL07], [HYSZ13] and [Her14] for details).

Using transfer matrix methods, the Krein-Rutman theory of positivity-preserving operators

and FK representation for the Ising model, [Her14] provides a region in the quadrant of

parameters β, µ > 0 where the innite-volume free energy has a limit, providing results on

convergence and asymptotic properties of the partition function and the Gibbs measure.

Thus, FK-Potts models, introduced by Fortuin and Kasteleyn (see [FK72]), prove that these

models have become an important tool in the study of phase transition for the Ising and

q-state Potts model.

The goal of this thesis is to use Krein-Rutman theory of positivity-preserving operators,

FK representation of the q-state Potts model on a xed triangulation and duality theory of

graph for study the q-state Potts model coupled to CDTs.

While recently much progress has been made in the development of analytical techniques

for CDT [JAZ07, JAZ08d], particularly random matrix models [JAZ08b, JAZ08a, JAZ08c],

and their application to multi-critical CDT [AGGS12, AZ12a, AZ12b], the causality con-

straints still makes it dicult to nd an analytical solution of the Ising model coupled to

CDT.

In this thesis we focus on study the q-state Potts model coupled to CDTs and is organised

as follows.

In Chapter 2 gives a summary of causal dynamical triangulations CDTs and we intro-

1.1 INTRODUCTION AND STATEMENT RESULTS 3

duced the transfer matrix formalism for pure CDTs. Also, we study asymptotic properties

of the partition function for pure CDTs. These properties will be used in next chapters.

In Chapter 3 we dene the annealed Ising model coupled to two-dimensional CDT and

develop a transfer matrix formalism. Spectral properties of the transfer matrix are rigorously

analysed by using the Krein-Rutman theorem [KR48] on operators preserving the cone

of positive functions. This yields results on convergence and asymptotic properties of the

partition function and the Gibbs measure and allows us to determine regions in the parameter

quarter-plane where the partition function converges. The main results of this chapter are

Lemma 3.2.1 and Theorem 3.2.2.

In Chapter 4 we use the Fortuin-Kasteleyn (FK) representation of quantum Ising models

via path integrals for determining a region in the quadrant of parameters β, µ > 0 where

the critical curve for the classical model can be located. In Section 4.1 we describe the

quantum Ising model. In Section 4.2, we give the FK representation of Ising model coupled

to CDTs via a path integral. This representation was originally derived in [MAC92] (see also

[Aiz94] and [Iof09]). Section 4.3 we present the main results of this chapter (Theorems 4.3.1

and 4.3.2). Section 4.4.1 and 4.4.2 contains the proof of Theorems 4.3.1 and 4.3.2. We also

provide lower and upper bounds for the innite-volume free energy. This chapter extends

results from Chapter 3 for the (annealed) Ising model coupled to two-dimensional causal

dynamical triangulations.

In Chapter 5. In Section 5.2, we introduce notation, dened the Potts model coupled

to CDTs and give a summary of the FK model, FK representation. Finally, we establish a

technical proposition of duality that will used in the next section. Section 5.3 contains the

proof of the rts main Theorem 5.1.1, and we nd a rst bounds for the critical curve. This

result will play a key role proof of the second main Theorem 5.1.2 of this chapter. In Section

5.4, using the High-T expansion for q-state Potts model, we prove Theorem 5.1.2.

Finally, Appendix A and B provide a review of trace class operators and Krein-Rutman

theory, used in Chapters 2 and 3.

Most of the novel results of this thesis have been published in research articles. In par-

ticular, the following chapters are based on the following articles:

• Chapter 2 and 3 on J.C. Hernández, Y. Suhov, A. Yambartsev, and S. Zohren, Bounds

on the critical line via transfer matrix methods for an Ising model coupled to causal

dynamical triangulations. J. Math. Phys. 54 063301 (2013).

• Chapter 4 on submitted paper, J. Cerda-Hernández, Critical region for an Ising model

coupled to causal dynamical triangulations. arxiv 1402.3251 (2014).

• Chapter 5 on preparation article, J. Cerda-Hernández, Duality relation for Potts model

coupled to causal dynamical triangulations (2014).

4 INTRODUCTION 1.1

Chapter 2

Two-dimensional causal dynamical

Triangulations

In this chapter we introduce causal dynamical triangulations (CDTs) as a discretization

of the partition function for two-dimensional quantum gravity. After giving a mathematical

denition of CDT we show some asymptotical properties of the partition function using

transfer matrix approach. These asymptotical properties will used in next sections.

2.1 Denitions

We will work with rooted causal dynamic triangulations of the cylinder CN = S × [0, N ],

N = 1, 2, . . . , which have N bonds (strips) S × [j, j+ 1]. Here S stands for a unit circle. The

denition of a causal triangulation starts by considering a connected graph G embedded

in CN with the property that all faces of G are triangles (using the convention that an

edge incident to the same face on two sides counts twice, see [SYZ13] for more details). A

triangulation t of CN is a pair formed by a graph G with the above propetry and the set F

of all its (triangular) faces: t = (G,F ).

Denition 2.1.1. A triangulation t of CN is called a causal triangulation (CT) if the

following conditions hold:

• each triangular face of t belongs to some strip S × [j, j + 1], j = 1, . . . , N − 1, and has

all vertices and exactly one edge on the boundary (S ×j)∪ (S ×j+ 1) of the stripS × [j, j + 1];

• if kj = kj(t) is the number of edges on S × j, then we have 0 < kj < ∞ for all

j = 0, 1, . . . , N − 1.

Denition 2.1.2. A triangulation t of CN is called rooted if it has a root. The root in the

triangulation t is represented by a triangular face t of t, called the root triangle, with an

anticlock-wise ordering on its vertices (x, y, z) where x and y belong to S1×0. The vertexx is identied as the root vertex and the (directed) edge from x to y as the root edge.

5

6 TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS 2.1

Denition 2.1.3. Two causal rooted triangulations of CN , say t = (G,F ) and t′ = (G′, F ′),

are equivalent if there exists a self-homeomorphism of CN which (i) transforms each slice

S1 × j, j = 0, . . . , N − 1 to itself and preserves its direction, (ii) induces an isomorphism

of the graphs G and G′ and a bijection between F and F ′, and (iii) takes the root of t to the

root of t′.

Let LTN and LT∞ denote the sets of causal triangulations on the nite cylinder CN and

innity cylinder C = S × [0,∞).

A triangulation t of CN is identied as a consistent sequence:

t = (t(0), t(1), . . . , t(N − 1)),

where t(i) is a causal triangulation of the strip S × [i, i + 1]. The latter means that each

t(i) is described by a partition of S × [i, i + 1] into triangles where each triangle has one

vertex on one of the slices S ×i, S ×i+ 1 and two on the other, together with the edge

joining these two vertices. The property of consistency means that each pair (t(i), t(i+ 1))

is consistent, i.e., every side of a triangle from t(i) lying in S × i+ 1 serves as a side of a

triangle from t(i+ 1), and vice versa.

The triangles forming the causal triangulation t(i) are denoted by t(i, j), 1 ≤ j ≤ n(t(i))

where, n(t(i)) stands for the number of triangles in the triangulation t(i). The enumeration

of these triangles starts with what we call the root triangle in t(i); it is determined recursively

as follows (see Figure 2.1(b)): First, we have the root triangle t(0, 1) in t(0) (see Denition

2.1.2). Take the vertex of the triangle t(0, 1) which lies on the slice S × 1 and denote it

by x′. This vertex is declared the root vertex for t(1). Next, the root edge for t(1) is the one

incident to x′ and lying on S×1, so that if y′ is its other end and z′ is the third vertex of the

corresponding triangle then x′, y′, z′ lists the three vertices anticlock-wise. Accordingly, the

triangle with the vertices x′, y′, z′ is called the root triangle for t(1). This construction can be

iterated, determining the root vertices, root edges and root triangles for t(i), 0 ≤ i ≤ N − 1.

It is convenient to introduce the notion of up" and down" triangles (see Figure 2.1(a)).

We call a triangle t ∈ t(i) an up-triangle if it has an edge on the slice S × i and a down-

triangle if it has an edge on the slice S×i+1. By Denition 2.1.1, every triangle is either of

type up or down. Let nup(t(i)) and ndo(t(i)) stand for the number of up- and down-triangles

in the triangulation t(i).

Note that for any edge lying on the slice S×i belongs to exactly two triangles: one up-

triangle from t(i) and one down-triangle from t(i− 1). This provides the following relation:

the number of triangles in the triangulation t is twice the total number of edges on the slices.

More precisely, let ni be the number of edges on slice S×i. Then, for any i = 0, 1, . . . , N−1,

n(t(i)) = nup(t(i)) + ndo(t(i)) = ni + ni+1, (2.1)

2.2 TRANSFER MATRIX FORMALISM FOR PURE CDTS 7

downtriangle up

triangle

S1×[ i, i +1]

root

(a) (b)

Figure 2.1: (a) A strip of a causal triangulation of S × [j, j + 1]. (b) Geometric representation of

a CDT with periodic spatial boundary condition.

implying thatN−1∑i=0

n(t(i)) = 2N−1∑i=0

ni. (2.2)

There is another useful property regarding the counting of triangulations. Let us x the

number of edges ni and ni+1 in the slices S × i and S × i + 1. The number of possible

rooted CTs of the slice S × [i, i+ 1] with ni up- and ni+1 down-triangles is equal to(ni + ni+1 − 1

ni − 1

)=

(n(t(i))− 1

nup(t(i))− 1

). (2.3)

2.2 Transfer matrix formalism for pure CDTs

We begin by discussing the case of pure causal dynamical triangulations, as was rst

introduced in [AL98] (see also [MYZ01] for a mathematically more rigorous account).

The partition function for rooted CTs in the cylinder CN with periodical spatial boundary

conditions (where t(0) is consistent with t(N − 1)) and for the value of the cosmological

constant µ is given by

ZN(µ) =∑t

e−µn(t) =∑

(t(0),...,t(N−1))

exp−µ

N−1∑i=0

n(t(i)). (2.4)

Using the properties (2.2) and (2.3) we can represent the partition function (2.4) in the

following way

ZN(µ) =∑

n0≥1,...,nN−1≥1

exp−2µ

N−1∑i=0

niN−1∏

i=0

(ni + ni+1 − 1

ni − 1

). (2.5)

8 TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS 2.2

Moreover, ZN(µ) admits a trace-related representation

ZN(µ) = tr(UN). (2.6)

This gives rise to a transfer matrix U = u(n, n′)n,n′=1,2,... describing the transition from

one spatial strip to the next one. It is an innite matrix with strictly positive entries

u(n, n′) =

(n+ n′ − 1

n− 1

)gn+n′ . (2.7)

For notational convenience we use the parameter g = e−µ (a single-triangle fugacity). The

entry u(n, n′) yields the number of possible triangluations of a single strip (say, S × [0, 1])

with n lower boundary edges (on S × 0) and n′ upper boundary edges (on S × 1). SeeFigure 5.2. The asymmetry in n and n′ is due to the fact that the lower boundary is marked

while the upper one is not. However, a symmetric transfer matrix U = u(n, n′) can be

introduced, associated with a strip where both boundaries are kept unmarked:

u(n, n′) = n−1u(n, n′). (2.8)

TheN -strip Gibbs distribution PN assigns the following probabilities to strings (n0, . . . , nN−1)

with the number of triangles ni ≥ 1 for all i = 0, . . . , N − 1:

PN,µ(n0, . . . , nN−1) =1

ZN(µ)exp−2µ

N−1∑i=0

niN−1∏

i=0

(ni + ni+1 − 1

ni − 1

). (2.9)

We state two lemmas featuring properties of matrix U :

Lemma 2.2.1. For any g > 0 the matrix U and its transpose UT have an eigenvalue

Λ = Λ(g) given by

Λ(g) =[(1−

√1− 4g2)/(2g)

]2

. (2.10)

The corresponding eigenvectors

φ = φ(n)n=1,2,... and φ∗ = φ∗(n)n=1,2,...

have entries

φ(n) = n(Λ(g)

)n, φ∗(n) = (Λ(g))n. (2.11)

Proof. A direct verication shows that∑n′

u(n, n′)n′Λn′(g) = nΛn+1(g) and∑n

Λn(g)u(n, n′) = Λn′+1(g).

(In fact, each of these relations implies the other.) See Theorem 1 in [MYZ01].

2.2 TRANSFER MATRIX FORMALISM FOR PURE CDTS 9

Lemma 2.2.2. For any xed n and any g < 1 (equivalently, µ > 0) one has∑n′

u(n, n′) =( g

1− g

)n(1− (1− g)n

). (2.12)

Proof. The proof again follows from a straightforward verication.

A transfer-matrix formalism of Statistical Mechanics predicts that, as N → ∞, the

partition function is governed by the largest eigenvalue Λ of the transfer matrix:

ZN(g) = tr UN ∼ ΛN (2.13)

We make this statement more precise in the statements of Lemma 2.2.3 and Theorem 2.2.1

below. Here the symbol `2 stands for the Hilbert space of square-summable complex se-

quences (innite-dimensional vectors) ψ = ψ(n)n=1,2,... equipped with the standard scalar

product 〈ψ′, ψ′′〉 =∑

n ψ′(n)ψ

′′(n). Accordingly, the matrices U and UT are treated as op-

erators in `2.

Lemma 2.2.3. For any g < 1/2 (equivalently µ > ln 2) the following statements hold true:

1. U and UT are bounded operators in `2 preserving the cone of positive vectors;

2. The sum∑

n,n′ u(n, n′) <∞. Consequently, U and UT have

tr(UUT

)= tr

(UTU

)<∞,

i.e., U and UT are Hilbert-Schmidt operators. Therefore, ∀ N ≥ 2, UN and(UT)N

are

trace-class operators.

3. The maximal eigenvalue Λ = Λ(g) of U in `2 is positive, coincides with the maximal

eigenvalue of UT and is given by Eqn (2.10). The corresponding eigenvectors φ, φ∗ ∈ `2

are unique up to multiplication by a constant factor and given in Eqn (2.11).

4. The following asymptotical formulas hold as N →∞:

1

ΛNtr(UN),

1

ΛNtr((UT)N

)→ 1,

and, ∀ vectors ψ′, ψ′′ ∈ `2,

1

ΛN〈ψ′, UNψ′′〉 = 〈ψ′, φ〉〈φ∗, ψ′′〉,

where the eigenvectors φ and φ∗ are normalized so that 〈φ, φ∗〉 = 1.

Theorem 2.2.1. For any g < 1/2 the following relation holds true:

limN→∞

1

Nlog ZN(g) = log Λ (2.14)

10 TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS 2.2

with Λ = Λ(g) given in (2.10). Further, the N-strip Gibbs measure PN,µ converges weakly to

a limiting measure Pµ which is represented by a positive recurrent Markov chain on Z+ =

1, 2, . . ., with the transition matrix P = P (n, n′)n=1,2,... and the invariant distribution π.

Here

P (n, n′) =u(n, n′)φ(n′)

Λφ(n)

and

π(n) =φ∗(n)φ(n)

〈φ∗, φ〉.

where φ(n) and φ∗(n) are as in (2.11).

Proof. The proof is a consequence of Lemma 2.2.1 and 2.2.3 and the Krein-Rutman theory

[KR48].

By Theorem 2.2.1, the measure on the set of innite triangulations LT∞ is then dened

as a weak limit

Pµ = limN→∞

PN .

The follow Theorem given the typical triangulation (typical behavior) under the limiting

measure Pµ.

Theorem 2.2.2 (See [MYZ01], [KY12]). The limit measure Pµ = limN→∞PN,µ exist for

all µ ≥ ln 2. Moreover, let nk be the number of vertices at k-th level in a triangulation t for

each k ≥ 0.

• For µ > ln 2 under the limiting measure Pµ the sequence nk is a positive recurrent

Markov chain.

• For µ = µcr = ln 2 the sequence nk is distributed as the branching process ξn with

geometric ospring distribution with parameter 1/2, conditioned to non-extinction at

innity.

Below we briey sketch the proof of the second part of Theorem 2.2.2, a deeper investi-

gation of related ideas will appear in [SYZ13].

Given a triangulation t ∈ LTN , dene the subgraph τ ⊂ t by taking, for each vertex

v ∈ t , the leftmost edge going from v downwards (see g. 1). The graph thus obtained is a

spanning forest of t , and moreover, if one associates with each vertex of τ it is height in t

then t can be completely reconstructed knowing τ . We call τ the tree parametrization of t.

For every vertex v ∈ τ denote by δv it is out-degree, i.e. the number of edges of τ going

from v upwards. Comparing the out-degrees in τ to the number of vertical edges in t ,

and comparing the latter to the total number of triangles n(t), it is not hard to obtain the

identity ∑v∈τ\S×N

(δv + 1) = n(t), (2.15)

2.2 TRANSFER MATRIX FORMALISM FOR PURE CDTS 11

Figure 2.2: Tree parametrization of a causal dynamical triangulation.

where the sum on the left runs over all vertices of τ except for the N -th level. Thus, under

the measure Pµcr the probability of a forest τ is proportional to

e−µcrn(t) =∏

v∈τ\S×N

(1

2

)δv+1

, (2.16)

which is exactly the probability to observe τ as a realization of a branching process with

ospring distribution Geom(1/2). After normalization we will obtain, on the left in (2.16),

the probability PN,µcr(τ) as dened by (2.9), an on the right the conditional probability to see

τ as a realization of the branching process ξ given ξN > 0. So quite naturally when N →∞the distribution of τ converges to the Galton-Watson tree, conditioned to non-extinction at

innity.

In particular it follows from Theorem 2.2.2 that

Pµcr(nk = m) = Pr(ξk = m|ξ∞ > 0) = mPr(ξk = m) (2.17)

Remark 2.2.1. The last equality in (2.17) means that the measure Pµcr on triangulations

can be considered as a Q-process dened by Athreya and Ney [AN72] for a critical Galton-

Watson branching process. Such a process is exactly a critical Galton-Watson tree conditioned

to survive forever.

In the supercritical case exp(−µ) < 1/2, we have the following asymptotical property of

the partition function

Proposition 2.2.1. In the supercritical case, exp(−µ) < 1/2, the nite volume partition

function ZN(µ) (dened in (2.4)) exist only if

µ > ln

(2 cos

π

N + 1

). (2.18)

Notice that, as N →∞ this region, where the partition function exists, become empty.

12 TWO-DIMENSIONAL CAUSAL DYNAMICAL TRIANGULATIONS 2.2

Remark 2.2.2. The inequality in (2.18) means that if µ < ln 2 then there exists N0 ∈ Nsuch that the partition function ZN(µ) = +∞ whenever N > N0. Moreover, the Gibbs

distribution PN,µ on triangulations with periodic boundary conditions cannot be dened by

using the standard formula with PN,µ as a normalising denominator, consequently, there is

no limiting probability measure Pµ.

Chapter 3

Transfer matrix formalism for Ising

model coupled to two-dimensional CDT

In this chapter we introduce a transfer matrix formalism for the (annealed) Ising model

coupled two-dimensional CDTs. Using the Krein-Rutman theory of positivity preserving

operators we study several properties of the emerging transfer matrix. In particular, we

determine regions in the quadrant of parameters β, µ > 0 where the innite-volume free

energy converges, yields results on the convergence and asymptotic properties of the partition

function and Gibbs measure. This is a rst approach for study the Ising model coupled two-

dimensional CDTs.

3.1 The model

Let t = (t(0), t(1), . . . , t(N − 1)) be a triangulation of CN , where t(i) is a causal trian-

gulation of the strip S × [i, i + 1]. The triangles forming the causal triangulation t(i) are

denoted by t(i, j), 1 ≤ j ≤ n(t(i)) where, n(t(i)) stands for the number of triangles in

the triangulation t(i). The enumeration of these triangles starts with what we call the root

triangle in t(i) (see Chapter 2).

Now, with any triangle from a triangulation t we associate a spin taking values ±1. An

N -strip conguration of spins is represented by a collection

σ = (σ(0),σ(1), . . . ,σ(N − 1))

where σ(i) = σ(t(i)) is a conguration of spins σ(i, j) over triangles t(i, j) forming a trian-

gulation t(i), 1 ≤ j ≤ n(t(i)). We will say that a single-strip conguration of spins σ(i) is

supported by a triangulation t(i) of strip S × [i, i+ 1]. We consider a usual (ferromagnetic)

Ising-type energy where two spins σ(i, j) and σ(i′, j′) interact if their supporting triangles

t(i, j), t(i′, j′) share a common edge; such triangles are called nearest neighbors, and this

property is reected in the notation 〈σ(i, j), σ(i′, j′)〉, where we require 0 ≤ i ≤ i′ ≤ N − 1.

Thus, in our model each spin has three neighbors. Moreover, a pair 〈σ(i, j), σ(i′, j′)〉 can

13

14 TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO

TWO-DIMENSIONAL CDT 3.1

only occur for i′− i ≤ 1 or i = 0, i′ = N − 1. Formally, the Hamiltonian of the model reads:

H(σ) = −∑

〈σ(i,j),σ(i′,j′)〉

σ(i, j)σ(i′, j′). (3.1)

We will use the following decomposition:

H(σ) =N−1∑i=0

H(σ(i)) +N−1∑i=0

V (σ(i),σ(i+ 1)), (3.2)

where we assume that σ(0) ≡ σ(N) (the periodic spatial boundary condition). Here H(σ(i))

represents the energy of the conguration σ(i):

H(σ(i)) = −∑

〈σ(i,j),σ(i,j′)〉

σ(i, j)σ(i, j′). (3.3)

Further, V (σ(i),σ(i+1)) is the energy of interaction between neighboring triangles belonging

to the adjacent strips S × [i, i+ 1] and S × [i+ 1, i+ 2]:

V (σ(i),σ(i+ 1)) = −∑

〈σ(i,j),σ(i+1,j′)〉

σ(i, j)σ(i+ 1, j′). (3.4)

The partition function for the (annealed) N -strip Ising model coupled to CDT, at the

inverse temperature β > 0 and for the cosmological constant µ, is given by

ΞN(µ, β) =∑

(t(0),...,t(N−1))

exp−µ

N−1∑i=0

n(t(i))

(3.5)

×∑

(σ(0),...,σ(N−1))

N−1∏i=0

exp−βH(σ(i))− βV (σ(i),σ(i+ 1))

.

Here n(t(i)) stands for the number of triangles in the triangulation t(i). Like before, the

formula

ΞN(µ, β) = tr KN (3.6)

gives rise to a transfer matrix K with entriesK((t,σ), (t′,σ′)) labelled by pairs (t,σ), (t′,σ′)

representing triangulations of a single strip (say, S × [0, 1]) and their supported spin cong-

urations which are positioned next to each other. Formally,

K((t,σ), (t′,σ′)) = 1t∼t′ exp−µ

2(n(t) + n(t′))

(3.7)

× exp−β

2

(H(σ) +H(σ′)

)− βV (σ,σ′)

.

As earlier, n(t) and n(t′) are the numbers of triangles in the triangulations t and t′. The

indicator 1t∼t′ means that the triangulations t, t′ have to be consistent with each other in the

3.1 THE MODEL 15

above sense: the number of down-triangles in t should equal the number of up-triangles in

t′, and an upper-marked edge in t should coincide with a lower-marked edge in triangulation

t′. It means that the pair (t, t′) forms a CDT for the strip S × [0, 2].

We would like to stress that the trace tr KN in (3.6) is understood as the matrix trace,

i.e., as the sum∑

t,σK(N)((t,σ), (t,σ)) of the diagonal entries K(N)((t,σ), (t,σ)) of the

matrix KN . (Indeed, in what follows, the notation tr is used for the matrix trace only.)

Our aim will be to verify that the matrix trace in (3.6) can be replaced with an operator

trace invoking the eigenvalues of K in a suitable linear space (see next section).

As before, we can introduce the N -strip Gibbs probability distribution associated with

formula (3.5):

PN((t(0),σ(0)), . . . , (t(N − 1),σ(N − 1))

)(3.8)

=1

Ξ(µ, β)

N−1∏i=0

exp−µn(t(i))− βH(σ(i))− βV (σ(i),σ(i+ 1))

.

Consider several special cases of interest.

The case β ≈ 0. This is the rst term of the so-called high temperature expansion [AAL99].

Here one has

Ξ(µ, 0) =∑

(t(0),...,t(N−1))

exp−µ

N−1∑i=0

n(t(i)) ∑

(σ(0),...,σ(N−1))

1

=∑

n0≥1,...,nN−1≥1

exp−2(µ− ln 2)

N−1∑i=0

niN−1∏

i=0

(ni + ni+1 − 1

ni − 1

)

= ZN(µ− ln 2); cf. (2.4).

The condition µ − ln 2 > ln 2 which guarantees properties listed in Lemma 2.2.3 and

Theorem 2.2.1 resuls in

µ > 2 ln 2. (3.9)

Thus, Eqn. (3.9) yields a sub-criticality condition when β = 0.

The case β ≈ ∞. Observe that for any triangulation t = (t(0), . . . , t(N − 1)) there are two

ground states: all spins +1 and all spins −1, with the overall energy equals minus three

half times the total number of triangles: −3/2∑N−1

i=0 n(t(i)). Discarding all other spin

congurations, we obtain that

Ξ(µ, β) > Ξ∗(µ, β)

16 TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO

TWO-DIMENSIONAL CDT 3.1

where

Ξ∗(µ, β) =∑

t(0),...,t(N−1)

2 exp(−µ+

3

2β)N−1∑i=0

n(t(i))

= 2∑

n0≥1,...,nN−1≥1

exp−2(µ− 3

2β)N−1∑i=0

ni(ni + ni+1 − 1

ni − 1

)

= 2ZN

(µ− 3

)

where exp

[3

2β∑i

n(t(i))

]is the energy of the (+)-conguration (or, equivalently,

the (−)-conguration). For β large, we can expect that Ξ(µ, β) ∼ Ξ∗(µ, β). Then the

critical inequality

µ− 3

2β > ln 2

yields

µ > ln 2 +3

2β. (3.10)

Equation (3.10) gives a necessary (and probably tight) criticality condition for the

Ising model under consideration for large values of β. A similar result was obtained in

[AAL99].

The case 0 < β <∞. Firstly, we note that for any xed triangulation t the energy of any

spin conguration σ on t will be bigger or equal than the energy of a pure conguration

(all +s or all −s):

H(σ) =∑j

H(σ(j)) +∑j

V (σ(j),σ(j + 1))

≥ −3

2#(of all triangles in t) = −3

N−1∑i=0

ni,

where ni is the number of edges in the ith level S × i, i = 0, 1 . . . , N − 1. Thus, for

any β > 0 the inequality Ξ(µ, β) < Ξ∗(µ, β) holds true, where

Ξ∗(µ, β) =∑

(t(0),...,t(N−1)

exp(−µ+

3

2β + ln 2

)N−1∑i=0

n(t(i))

=∑

n0≥1,...,nN−1≥1

exp−2(µ− 3

2β − ln 2

)N−1∑i=0

ni

= ZN(µ− 3

2β − ln 2

).

3.2 THE TRANSFER-MATRIX K AND ITS POWERS KN 17

Hence, the inequality

µ− 3

2β − ln 2 > ln 2 or µ > 2 ln 2 +

3

2β (3.11)

provides a sucient condition for subcriticality of the Ising model under consideration.

3.2 The transfer-matrix K and its powers KN

The main results of this chapter are summarized in Lemma 3.2.1 and Theorems 3.2.1

and 3.2.2 below.

Let us start with a statement (see Proposition 3.2.1 below) which merely re-phrases

standard denitions and explains our interest in the matrices K, KT, KTK, KKT and their

powers. Cf. Denition 2.2.2 on p.83, Denition 2.4.1 on p.101, Lemma 2.3.1 on p.85 and

Theorem 3.3.13 on p.139 in [Rin71]). See Appendix A for a short review.

We treat the transfer-matrix K and its transpose KT as linear operators in the Hilbert

space `2T−C (the subscript T-C refers to triangulations and spin-congurations). The space

`2T−C is formed by functions ψ = ψ(t,σ) with the argument (t,σ) running over single-strip

triangulations and supported congurations of spins, with the scalar product 〈ψ′,ψ′′〉T−C =∑t,σ ψ

′(t,σ)ψ′′(t,σ) and the induced norm ‖ψ‖T−C. The action of K in `2T−C, in the basis

formed by Dirac's delta-vectors δ(t,σ), is determined by

(Kψ

)(t,σ) =

∑t′,σ′

K((t,σ), (t′,σ′))ψ(t′,σ′); (3.12)

in following we use the notation K, KT, etc., for the matrices and the corresponding operators

in `2T−C. Accordingly, the symbols ‖K‖T−C, ‖KT‖T−C etc. refer to norms in `2

T−C.

Given n = 1, 2, . . ., suppose that the operator Kn (respectively,(KT)n) is of trace class

(see denition in Appendix A). Then the following series absolutely converges:

∑j

Λ(n)j

(respectively,

∑j

Λ∗(n)j

), (3.13)

where Λ(n)j (Λ∗

(n)j ) runs through the eigenvalues of Kn ((KT)n), counted with their multi-

plicities. In this case the sum (3.13) is called the operator trace of Kn (respectively, (KT)n)

in `2T−C. We adopt an agreement that the eigenvalues in (3.13) are listed in the decreasing

order of their moduli, beginning with Λ(n)0 (Λ∗

(n)0 ).

Set |Kn| =√

(KT)n Kn and∣∣(KT

)n∣∣ =√

Kn (KT)n.

18 TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO

TWO-DIMENSIONAL CDT 3.2

Proposition 3.2.1. For any positve integer r, the following inequalities are equivalent:

tr((

Kr(KT)r)

= tr((

KT)r

Kr)<∞ and

tr|K2r| = tr|(KT)2r| <∞.

(3.14)

Moreover, each of the inequalities in (3.14) implies that ∀ N ≥ 2r, the operators KN

and (KT)N are of trace class in `2T−C. Hence, for N ≥ 2r, the matrix traces tr

(KN)and

tr((KT)N) are nite and coincide with the corresponding operator traces in `2T−C.

Theorem 3.2.1. Suppose that the condition (3.14) is satised with r = 1. Then the following

properties of transfer matrix K are fulllled.

1. The square K2 and its transpose (KT)2 are trace-class operators in `2T−C.

2. K and KT have a common eigenvalue, Λ = Λ0(β, µ) > 0 such that the norms

‖K‖T−C = ‖KT‖T−C = Λ. Furthermore, K2 and (KT)2 have the common eigenvalue

Λ2 = Λ(2)0 = Λ∗

(2)0 such that the norms ‖K2‖T−C = ‖(KT)2‖T−C = Λ2 .

3. Λ is a simple eigenvalue of K and KT, i.e., the corresponding eigenvectors φ =

φ(t,σ) and φ∗ = φ∗(t,σ) are unique up to multiplicative constants. Moreover,

φ and φT can be made strictly positive: φ(t,σ),φT(t,σ) > 0 ∀ (t,σ). Furthermore,

Λ is separated from the remaining singular values and the remaining eigenvalues of K

and KT by a positive gap. The same is true for Λ2 and K2 and(KT)2.

Proof of Theorem 3.2.1. Because the entries K((t,σ), (t′,σ′)) are non-negative, the con-

dition (3.14) with r = 1 means that∑(t,σ),(t′,σ′)

K2((t,σ), (t′,σ′)) <∞, (3.15)

that is, K and KT are Hilbert-Schmidt operators. It means that the operator KKT has an

orthonormal basis of eigenvectors and the series of squares of its eigenvalues (counted with

multiplicities) converges and gives the trace trT−C(KKT). Consequently, the operators K

and KT are bounded (and even completely bounded) and K2 and (KT)2 are of trace class.

The latter fact means that the matrix trace of the operator K2 coincides with its operator

trace in `2T−C, and the same is true of (KT)2. In addition, the operator K2 has the property

that its matrix entries K(2)((t,σ), (t′,σ′)) are strictly positive:

K(2)((t,σ), (t′,σ′)) =∑(t,σ)

K((t,σ), (t, σ))K((t, σ), (t′,σ′)) > 0. (3.16)

The KreinRutman theory (see [KR48], Proposition VII′ or Appendix B) guarantees that

both K and KT have a maximal eigenvalue Λ that is positive and non-degenerate, or simple.

3.2 THE TRANSFER-MATRIX K AND ITS POWERS KN 19

That is, the eigenvector φ of K and the eigenvector φ∗ of KT corresponding with Λ are

unique up to multiplication by a constant, and all entries φ(t,σ) and φ∗(t,σ) are non-

zero and have the same sign. In other words, the entries φ(t,σ) and φ∗(t,σ) can be made

positive. The spectral gaps are also consequences of the above properties.

Set:

λ(µ, β) = c2 (m2 + 1) (cosh 2β)

(1 +

√1− 1

(cosh 2β)2

(m2 − 1)2

(m2 + 1)2

)(3.17)

where c and m are determined by

c =exp(β − µ)

e2β(1− exp(β − µ))2 − e−2µ(3.18)

m = e2β + (1− e4β) exp (−(β + µ)). (3.19)

Lemma 3.2.1. For any β, µ > 0 such that

λ(µ, β) < 1, (3.20)

the condition (3.14) is satised for r = 1:

tr(KKT) = tr(KTK) <∞ and tr|K2| = tr|(KT)2| <∞, (3.21)

implying the assertions of Proposition 3.2.1 and Theorem 3.2.1. Moreover, the condition

(3.14) implies (3.20)

Proof of Lemma 3.2.1. By denition the trace (3.21) we need to calculate the series

tr(KTK) =∑(t,σ)

KTK((t,σ), (t,σ))

=∑

(t,σ),(t′,σ′)K((t,σ), (t′,σ′))K((t,σ), (t′,σ′))

=∑

(t,σ),(t′,σ′)K2((t,σ), (t′,σ′)). (3.22)

A single-strip triangulation t consists of up- and down-triangles. Accordingly, it is con-

venient to employ new labels for spins: if a triangle t(l) is an lth up-triangle then we denote

it by tlup; the corresponding spin σ(j) will be denoted by σlup. Similarly, if t(j) is an lth

down-triangle then we denote it by tldo; the spin σ(j) will be denoted by σldo. Consequently,

the triangulation t and its supported spin-conguration σ are represented as

t := (tup, tdo) and σ := (σup,σdo).

20 TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO

TWO-DIMENSIONAL CDT 3.2

Here

tup = (t1up, . . . , tnup), tdo = (t1do, . . . , t

mdo),

and

σup = (σ1up, . . . , σ

nup), σdo = (σ1

do, . . . , σmdo),

assuming that the supporting single-strip triangulation t contains n up-triangles and m

down-triangles. (The actual order of up- and down-triangles and supported spins does not

matter.)

The same can be done for the pair (t′,σ′) as illustrated in (3.22). Let recall that the

triangulations t and t′ are consistent (t ∼ t′) i number of the down-triangles in t equals

that of up-triangles in t′.

To calculate the sum (3.22) we divide the summation over (t′,σ′) into a summation over

(t′up,σ′up) and (t′do,σ

′do). Firstly, x a pair (t′up,σ

′up) and make the sum over (t′do,σ

′do). Note

that the term V ((t,σ), (t′,σ′)) depends only on σdo and σ′up. Consequently,

∑t′do,σ′do

K2((t,σ), (t′,σ′)) (3.23)

= e−βH(σ)e−2βV ((t,σ),(t′,σ′))e−µn(t)∑

(t′do,σ′do)

e−βH(σ′)e−µn(t′).

The sum in the right-hand side of (3.23) can be represented in a matrix form. Denote by

e±1 the standard spin-1/2 unit vectors in R2:

e+1 =

1

0

and e−1 =

0

1

.

Next, let us introduce a 2× 2 matrix T where

T = e−µ

eβ e−β

e−β eβ

:=

t++ t+−

t−+ t−−

. (3.24)

Denote by n(i), i = 1, . . . , nup(t′) the number of down-triangles in t′ which are between the

3.2 THE TRANSFER-MATRIX K AND ITS POWERS KN 21

ith and (i+ 1)th up-triangles in t′. Let nup(t′) = k then

∑t′do,σ′do

e−βH(σ′)e−µn(t′) =∑

n(i)≥0:∑i n(i)≥1

k∏l=1

(eTσ′lup

T n(l)+1eσ′l+1up

)

=k∏l=1

(eTσ′lup

Meσ′l+1up

)−

k∏l=1

(eTσ′lup

Teσ′l+1up

)(3.25)

where the matrix M is the sum of the geometric progression

M =∞∑n=1

T n :=

m++ m+−

m−+ m−−

. (3.26)

Using the same procedure we can obtain the sum over all up-triangles into the triangulation

t. The only dierence is the existence of marked up-triangle in the strip: let as before

nup(t′) = ndo(t) = k then

∑tup,σup

e−βH(σ)e−µn(t) =k−1∏l=1

(eTσlupMeσl+1

up

)(eTσkupM2eσ1

up

)(3.27)

See Figure 3.1 for illustration of these calculations (3.25) and (3.27). Further, supposing the

existence of the matrix M and using (3.25) and (3.27) we obtain the following:∑tup,σup

∑t′do,σ′do

K2((t,σ), (t′,σ′)) = e−2βV ((tdo,σdo),(t′up,σ′up))

×∑

tup,σup

e−βH(σ)e−µn(t)∑

(t′do,σ′do)

e−βH(σ′)e−µn(t′)

= e−2βV ((tdo,σdo),(t′up,σ′up))

×[ k∏l=1

(eTσ′lup

Meσ′l+1up

) k−1∏l=1

(eTσldoMeσl+1

do

)(eTσkupM2eσ1

up

)−

k∏l=1

(eTσ′lup

Teσ′l+1up

) k−1∏l=1

(eTσldoMeσl+1

do

)(eTσkupM2eσ1

up

)]. (3.28)

Necessary and sucient condition for the convergence of the matrix series for M is that

the maximal eigenvalue of matrix T is less then 1. The eigenvalues of T are

λ± = e(β−µ) ± e−(β+µ), (3.29)

22 TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO

TWO-DIMENSIONAL CDT 3.2

t ',σ '( )

t,σ( )

σ 'up1

σ do1

σ 'up2

σ do2

σ 'up3

σ do3

mσ 'up1 σ 'up

2 mσ 'up2 σ 'up

3

mσ 'up3 σ 'up

1

mσ do1 σ do

2 mσ do2 σ do

3 mσ do3 σ do

1(2)

Figure 3.1: Illustration of the calculates (3.25) and (3.27).

and the above condition means that λ+ < 1 or, equivalently,

µ > ln(2cosh(β)

). (3.30)

Under this condition (3.30), the matrix M is calculated explicitly:

M =e(β−µ)

e2β(1− e(β−µ))2 − e−2µ

×

e2β + (1− e4β)e−(β+µ) 1

1 e2β + (1− e4β)e−(β+µ)

.

(3.31)

We are now in a position to calculate the sum in (3.22). To this end, we again represent

it through the product of transfer matrices. Pictorially, we express the above sum as the

partition function of a one-dimensional Ising-type model where states are pairs of spins

(σldo, σlup) and the interaction is via the matrix T between the members of the pair and via

3.2 THE TRANSFER-MATRIX K AND ITS POWERS KN 23

matrix M between neighboring pairs. More precisely, dene the following 4× 4 matrices:

Q =

e2βm++m++ m++m+− m+−m++ e2βm+−m+−

m++m−+ e−2βm++m−− e−2βm+−m−+ m+−m−−

m−+m++ e−2βm−+m+− e−2βm−−m++ m−−m+−

e2βm−+m−+ m−+m−− m−−m−+ e2βm−−m−−

(3.32)

Qm =

e2βm++m(2)++ m++m

(2)+− m+−m

(2)++ e2βm+−m

(2)+−

m++m(2)−+ e−2βm++m

(2)++ e−2βm+−m

(2)−+ m+−m

(2)−−

m−+m(2)++ e−2βm−+m

(2)+− e−2βm−−m

(2)++ m++m

(2)++

e2βm−+m(2)−+ m−+m

(2)−− m−−m

(2)−+ e2βm−−m

(2)−−

(3.33)

Qt =

e2βt++m++ t++m+− t+−m++ e2βt+−m+−

t++m−+ e−2βt++m−− e−2βt+−m−+ t+−m−−

t−+m++ e−2βt−+m+− e−2βt−−m++ t−−m+−

e2βt−+m−+ t−+m−− t−−m−+ e2βt−−m−−

(3.34)

Qtm =

e2βt++m(2)++ t++m

(2)+− t+−m

(2)++ e2βt+−m

(2)+−

t++m(2)−+ e−2βt++m

(2)++ e−2βt+−m

(2)−+ t+−m

(2)−−

t−+m(2)++ e−2βt−+m

(2)+− e−2βt−−m

(2)++ t++m

(2)++

e2βt−+m(2)−+ t−+m

(2)−− t−−m

(2)−+ e2βt−−m

(2)−−

(3.35)

24 TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO

TWO-DIMENSIONAL CDT 3.2

where mij,m(2)ij and ti,j (i, j ∈ −,+) are elements of the matrices M,M2, and T respec-

tively.

Now for the sum under consideration (3.22) we obtain using representation (5.53)∑(t,σ),(t′,σ′)

K2((t,σ), (t′,σ′)) =∑

(tdo,σdo),(t′up,σ′up)

e−2βV ((tdo,σdo),(t′up,σ′up))

×

[k∏l=1

(eTσ′lup

Meσ′l+1up

) k−1∏l=1

(eTσldoMeσl+1

do

)(eTσkupM2eσ1

up

)−

k∏l=1

(eTσ′lup

Teσ′l+1up

) k−1∏l=1

(eTσldoMeσl+1

do

)(eTσkupM2eσ1

up

)]

= tr(( ∞∑

k=0

Qk)Qm

)− tr

(( ∞∑k=1

Qkt

)Qtm

). (3.36)

By the construction the matrix Q is greater then Qt elementwise. Thus the eigenvalue of

matrixQ is greater than the eigenvalue of the matrixQt (it follows from the Perron-Frobenius

theorem). Therefore the necessary and sucient condition for the convergence in (3.22) is

that the largest eigenvalue of Q is less than 1. It is possible to calculate its eigenvalue

analytically. In order to express the eigenvalues of Q it is convinient to use notations (3.18)

and (3.19). In this notations the matrix M , i.e. (3.31), is represented as following

M = c

m 1

1 m

.

The equations for the eigenvalues of Q are:

λ1 = c2eβ(m2 − 1)

λ2 = c2e−β(m2 − 1)

λ3 = c2(m2 + 1)(cosh β)

(1−

√1− (m2 − 1)2

(cosh β)2(m2 + 1)2

)

λ4 = c2(m2 + 1)(cosh β)

(1 +

√1− (m2 − 1)2

(cosh β)2(m2 + 1)2

)

A straightforward inspection conrms that the largest eigenvalue is given by λ4. The con-

dition λ4 < 1 coincides with (3.20). Finally, using matrices Q,Qm (see formulas (3.32) and

(3.33)) with positive entries and of size 4× 4, we have the following representation of 3.22

tr(KKT) = tr((∑

k≥1

Qk)Qm

)+ . . . .

3.3 DISCUSSION AND OUTLOOK 25

The convergence of the matrix series∑

k≥1Qk is equivalent to the condition that the maximal

eigenvalue of the matrix Q is less then 1. This is exactly the condition (3.20). This completes

the proof of Lemma 3.2.1.

Theorem 3.2.2. Under condition (3.20), the following limit holds:

limN→∞

1

Nlog ΞN(β, µ) = log Λ. (3.37)

Moreover, as N → ∞, the N-strip Gibbs measure PN (see Eqn (5.9)) converges weakly to

a limiting probability distribution P that is represented by a positive recurrent Markov chain

with states (t,σ), the transition matrix

P = P ((t,σ), (t′,σ′)) and the invariant distribution π = π(t,σ) where

P ((t,σ), (t′,σ′)) =K((t,σ), (t′,σ′))φ(t′,σ′)

Λφ(t,σ)

π(t,σ) = φ(t,σ)φT(t,σ)/⟨φ,φT

⟩T−C

with the norm∥∥φ∥∥2

T−C=∑

t,σ φ(t,σ)2.

Proof of Theorem 3.2.2. The spectral gap for K implies that ∀ ψ ∈ `2T−C, we have the

convergence

limN→∞

1

ΛNKNψ = (〈ψ,φ〉T−C)φ

in the norm of space `2T−C. Moreover, let Π denote the operator of projection to the subspace

spanned by the eigenvectors of K dierent from φ. Then

1

Λ‖ΠKP‖T−C < 1 =⇒ lim

N→∞

1

ΛN

∥∥∥(ΠKP)N∥∥∥

T−C= 0.

In turn, this implies that

1

Nlog ΞN(µ, β) =

1

Nlog trT−CKN → log Λ.

Convergence of the Gibbs measure PN follows as a corollary.

3.3 Discussion and outlook

This chapter makes a step towards determining the subcriticality domain for an Ising-

type model coupled to two-dimensional causal dynamical triangulations (CDT). In doing

so we employ transfer-matrix techniques and in particular the Krein-Rutman theorem. We

complement the discussion of the previous sections with the following two concluding re-

marks:

26 TRANSFER MATRIX FORMALISM FOR ISING MODEL COUPLED TO

TWO-DIMENSIONAL CDT 3.3

Remark 1. It is instructive to summarise the logical structure of the argument establishing

Lemma 3.2.1 and Theorems 3.2.1 and 3.2.2:

• First, (3.21) holds i condition (3.20) holds: see the proof of Lemma 3.2.1.

• Next, (3.21) implies that K is a HilbertSchmidt operator and K2 is a trace class

operator in `2T−C.

• The last fact, together with the property of positivity (3.16), allow us to use the Krein

Rutman theory, deriving all assertions of Theorems 3.2.1 and 3.2.2.

On the other hand, if (3.20) fails (and therefore (3.21) fails), it does not necessarily mean

that the assertions Theorems 3.2.1 and 3.2.2 fail. In other words, we do not claim that the

boundary of the domain of parameters β and µ where the model exhibits uncritical behavior

is given by Eqn. (3.20). Moreover, Figure 3.2 shows the result of a numerical calculation

indicating that the condition (3.20) is worse than (3.11) for (moderately) large values of β.

An apparent condition closer to necessity is the pair of inequalities (3.14) for some (pos-

sibly) large r. This issue needs a further study.

Remark 2. Physical considerations suggest that the critical curve in the (β, g) quarter-

plane would have some predictable patterns of behavior: as a function of β, it would decay

and exhibit a rst-order singularity at a unique point β = βcr ∈ (0,∞).

A plausible conjecture is that the boundary of the critical domain coincides with the locus

of points (β, µ) where Λ looses either the property of positivity or the property of being a

simple eigenvalue. This direction also requires further research.

𝛽 ≈ 0

𝜇

𝛽

𝜆 ≤1

𝜆 ≤1

Figure 3.2: λQ = λ and λT are the maximal eigenvalues of the matrix Q and a related matrix Trespectively. The area above the black curve is where the condition (3.20) holds true.

Chapter 4

FK representation for the Ising model

coupled to CDT

This chapter extends results from before chapter for the (annealed) classical Ising model

coupled to two-dimensional causal dynamical triangulations. Using the Fortuin-Kasteleyn

(FK) representation of quantum Ising models via path integrals, we determine a region in

the quadrant of parameters β, µ > 0 where the critical curve for the classical model can be

located. In particular, we determine a region where the innite-volume Gibbs measure exists

and it is unique, and a region where the nite-volume Gibbs measure has no weak limit (in

fact, does not exist if the volume is large enough). We also provide lower and upper bounds

for the innite-volume free energy.

FK models were introduced by Fortuin and Kasteleyn (see [FK72]). These models have

become an important tool in the study of phase transition for the Ising and Potts model. The

goal of this chapter is to introduce the FK representation of a quantum Ising model coupled to

CDTs via a path integral (see [Aiz94], [Iof09] for an overview), and use this representation for

obtain information of the critical curve. The aforementioned FK representation uses a family

of Poisson point processes and the Lie-Trotter product formula to interpret exponential

sums of operators as random operator products. This representation was originally derived

in [Aiz94].

4.1 The quantum Ising model

In this section we write the classical partition function, over a given triangulation, by

using ingredients of the quantum Ising model.

Henceforth, for simplicity in notation and exposure of the following chapter, we shall

denote a triangle of any triangulation t doing without put the indices i, j as was done in

previous chapter. Thus, in this chapter, the Hamiltonian the (annealed) model is written as

follow

H(σ) = −∑〈t,t′〉

σ(t)σ(t′). (4.1)

27

28 FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT 4.1

Here, 〈t, t′〉 stands that triangles t, t′ have a common edge. These triangles are called nearest

neighbor.

Let t = (t(0), t(1), . . . , t(N − 1)) be a causal triangulation of CN with periodical spa-

tial boundary condition (see Figure 3.1 (b)). Let ∆(t) denote the set of triangles of the

triangulation t.

We dene Ω(t) to be set of all spin congurations supported by the triangles of t, i.e.,

Ω(t) = −1,+1∆(t). Let Zβ,tN be the partition function of the Ising model on the CDT t,

at inverse temperature β > 0

Zβ,tN =∑σ∈Ω(t)

exp−βH(σ), (4.2)

where H(σ) represents the energy of conguration σ ∈ Ω(t), dened by the formula (4.1).

The quantum Ising model on a causal triangulation t is dened as follows.

Let

σz =

1 0

0 −1

(4.3)

be the Pauli matrix with their corresponding eigenvectors

φ+1 =

1

0

and φ−1 =

0

1

. (4.4)

In the quantum lenguage spins values ±1 are understood as eigenvalues of Pauli matrix.

Notice that σzφν = νφν for ν = ±1.

To each triangle t ∈ ∆(t) we associate a spin taking values φ+1 and φ−1. Thus, the space

of all such spin congurations on t is dene as the real vector space Xt =⊗

t∈∆(t) R2, where⊗stands for the tensor product. Notice that Xt is a real vector space of dimension 2 to the

n(t) power: dim(Xt) = 2n(t).

For each classical conguration σ ∈ Ω(t) we associate the quantum conguration as

tensor products

φσ := ⊗t∈∆(t)φσ(t),

where σ(t) is the spin supported by the triangle t ∈ ∆(t). Notice that there is a one-

one correspondence between Ω(t) and the collection φσσ∈Ω(t). Moreover, the collection of

quantum congurations is a complete orthonormal basis of Xt with respect to the following

4.2 FK REPRESENTATION FOR ISING MODEL COUPLED TO CDT 29

scalar product

〈φσ|φσ′〉 :=∏t∈∆(t)

(φσ(t),φσ′(t)

)2,

where (·, ·)2 is the usual scalar product of R2. With each triangle t ∈ ∆(t) we associate a

linear self-adjoint operator σzt : Xt → Xt which acts as a copy of Pauli matrix σz on the

coordinate of φσ associated to the triangle t of t. That is, for each σ ∈ Ωt,

σztφσ = φσ(t1) ⊗ · · · ⊗(σzφσ(t)

)⊗ · · · = σ(t)φσ. (4.5)

Note that operators σzt , σzt′ commute, and satises

σzt σzt′φσ = σ(t)σ(t′)φσ. (4.6)

The Hamiltonian Ht of the quantum Ising model is a linear self-adjoint operator dened on

Xt:

Ht = −∑〈t,t′〉

σzt σzt′ , (4.7)

where two operators σzt and σzt′ interact if their supporting triangles t, t′ ∈ ∆(t) are nearest

neighbors.

Note that Htφσ = H(σ)φσ. In other words, Ht is a diagonal in the φσ basis, and

corresponding eigenvalues being equal to values of the classical Ising Hamiltonian on cong-

urations σ. This allows write the classical partition function Zβ,tN for Ising model, at inverse

temperature β > 0 associated with triangulation t, as follows

Zβ,tN =∑σ∈Ω(t)

exp−βH(σ) =∑σ∈Ω(t)

〈φσ|e−βHt|φσ〉 = tr(e−βHt

). (4.8)

Finally, using the quantum representation (4.8), the partition function for the N -strip Ising

model coupled to CDT, at the inverse temperature β > 0 and for the cosmological constant

µ, can be written as follows

ΞN(β, µ) =∑t

e−µn(t)tr(e−βHt

). (4.9)

4.2 FK representation for Ising model coupled to CDT

In order to calculate tr(e−βHt

)in (4.9) we will use the FK representation for the Ising

model via path integrals, see [Aiz94, Iof09]. By representation (4.9), the trace tr(e−βHt

)may be expressed in terms of a type of path integral with respect to the continuous random-

cluster model on ∆(t)×[0, β] for any Lorentzian triangulation t (see Proposition 4.2.1 below).

For any pair t, t′ ∈ ∆(t) of nearest neighbor triangles, we associate a Poisson process

30 FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT 4.2

ξ〈t,t′〉(s) on the time interval [0, β] with intensity 2. We refer to the process ξ〈t,t′〉 as process

of arrivals of operator K〈t,t′〉 on the interval [0, β], where

K〈t,t′〉 =I + σzt σ

zt′

2. (4.10)

Let ξ be the collection of independent Poisson processes ξ〈t,t′〉 : ξ(s) := ξ〈t,t′〉(s)〈t,t′〉∈Et ,where Et is the set of all pairs of neighbor triangles: Et := 〈t, t′〉 : t, t′ ∈ ∆(t).

Let Pβ,t denote the probability measure associated with the family of Poisson process

ξ. We shall abuse notation by using ξ to denote a realization of process of arrivals ξ(s),

s ∈ [0, β]. By independence there are no simultaneous arrivals Pβ,t-a.s. Thus, a realization ξ

of process of arrivals can be represented by a collection of arrival times sii=1,...,Nξ contained

in [0, β] and its corresponding arrival types L(si) ∈ Et, ξ ≡ si, L(si)i=1,...,Nξ , where Nξ is

the total number of arrivals during the time [0, β].

With a xed realization ξ we associated a family of all possible piecewise constant right-

continuous functions ψξ = ϕ : [0, β] → φσ, having jumps only at arrival times of

ξ. Since Xt is nite dimensional and there are Pβ,t-a.s. nite number of arrivals, we have

|ψξ| <∞, Pβ,t-a.s., where |ψξ| is the total number of functions in the set ψξ.

For each arrival time s of a realization ξ corresponding a unique arrival type L(s) a.s.

Suppose that L(s) = 〈t, t′〉 for t, t′ ∈ ∆(t) nearest neighbor, then KL(s) = K〈t,t′〉 : Xt → Xt.

Let ϕ ∈ ψξ, and denote ϕ(s−) = limt→s− ϕ(t). Notice that the function ϕ ∈ ψξ can be

continuous or not at each arrival time s (see Figure 4.1 below).

arrival ofoperator

arrival ofoperator

Figure 4.1: A trajectory sample associated with a realization ξ = sii=1,...,n. Each trajectory

ϕ ∈ ψξ can be continuous or not at each arrival time s. In this case, at arrival time sk−1 the

trajectory ϕ do not have jump, and at arrival time sk the trajectory ϕ have a jump.

Using the before notation, we have the following proposition.

4.2 FK REPRESENTATION FOR ISING MODEL COUPLED TO CDT 31

Proposition 4.2.1. The matrix elements of the linear operator e−βHt with respect to the

basis φσ are given by

⟨φσ|e−βHt|φσ′

⟩= exp

3

2βn(t)

∫Pβ,t(dξ)

∑ϕ∈ψξ

ϕ(0)=φσ ,ϕ(β)=φσ′

∏s∈ξ

〈ϕ(s−)|KL(s)|ϕ(s)〉, (4.11)

for all t ∈ LTN .

Formula (4.11) was proved in [Aiz94] and [Iof09] for any general nite graph.

With any realization ξ we associate a graph Gξ = (∆ξ, Eξ), where the set of vertices is

∆ξ = ∆(t) and the set of edges Eξ ⊆ Et is dened by following rule: an edge e = 〈t, t′〉 ∈ Etbelong to Eξ if and only if there exist a arrival time s such that the corresponding arrival

type L(s) is 〈t, t′〉 into the realization ξ.

We say that two triangulations t and t′ are connected, denoted by t ↔ t′, if and only if

there exist a path inGξ connecting t and t′. For any t ∈ ∆(t), we suppose that t↔ t. A subset

C ⊆ ∆(t) is called a cluster (maximal connected component) if for any t, t′ ∈ C then t↔ t′,

and t = t′ for any t ∈ C and t′ /∈ C (see Figure 5.1 below). Thus, any realization ξ of the

Poisson process splits ∆(t) into the disjoint union of maximal connected components, i.e., for

any realization ξ there exists k = k(ξ) ∈ 1, . . . , n(t) and sets C1 = C1 . . . , Ck = Ck(ξ) ⊆ ∆(t)

such that

∆(t) =

k(ξ)⋃i=1

Ci,

and Ci ∩ Cj = ∅ for i 6= j, Here k(ξ) is the number of clusters dened by the relation ↔.

Additionally, we dene the cluster Ct of a triangle t by Ct = t′ ∈ ∆(t) : t↔ t′.

Time

Figure 4.2: In this gure, we show the Cluster Ct of a triangle t, and a graphic representation of

relation t↔ t′, where ↔ on right side in the gure, represent arrival times.

Let σ,σ′ ∈ Ω(t) be two congurations and let φσ,φσ′ be the corresponding quantum

32 FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT 4.3

conguration. Then, for any 〈t, t′〉 ∈ Et

〈φσ|K〈t,t′〉|φσ′〉 = δσ=σ′δσ(t)=σ(t′). (4.12)

Relation (4.12) implies that, for any realization ξ only constant functions of ψξ contribute to

the sum inside the integral (4.11). Additionally, an arrival of K〈t,t′〉 at arrival time s ∈ [0, β]

imposes an additional condition σ(t) = σ(t′) for contribute to the sum in (4.11). Dene

Ω(t, ξ) = σ ∈ Ω(t) : σ has same sign in each cluster Ci.

Notice that |Ω(t, ξ)| = 2k(ξ), and

∑ϕ∈ψξ

ϕ(0)=ϕ(β)=φσ

∏s∈ξ

〈ϕ(s−)|KL(s)|ϕ(s)〉 =

1 if σ ∈ Ω(t, ξ)

0 if σ /∈ Ω(t, ξ)

(4.13)

As an elementary consequence of (4.13) the following representation for partition function

Zβ,tN holds.

Proposition 4.2.2. Let t ∈ LTN and β > 0. We have that

Zβ,tN = tr(e−βHt

)= exp

3

2βn(t)

∫2k(ξ)Pβ,t(dξ). (4.14)

Proof. The proof is consequence of Proposition 4.2.1 and equation (4.13).

Using the N -strip Gibbs probability distribution PN,µ (introduced in Eqn (2.9)) for pure

CDTs with periodical boundary condition, and substituting (4.14) on the right-hand side

of (4.9) we obtain the FK representation of partition function for the N -strip Ising model

coupled to CDTs, at inverse temperature β > 0 and for the cosmological constant µ

ΞN(β, µ) = ZN(r)∑

t∈LTN

∫2k(ξ)Pβ,t(dξ)

PN,r(t), (4.15)

where r = µ− 32β and ZN(·) is dened by (2.4).

4.3 The main results

This section contains the statement of the main theorems of the present chapter.

4.3 THE MAIN RESULTS 33

Understanding by critical curve of the model the boundary of the domain of parameters β

and µ where the model exhibits subcritical behavior, this chapter makes a rigorous derivation

of a subcriticality domain for an Ising model coupled to two-dimensional CDTs, and we nd

a domain where the tipical innite-volume Gibbs measure there no exists. In Figure 4.3, we

show a region where the critical curve of the model should be located. Formally, we dene

the critical curve as follow: We denote by Gβ,µ the set of Gibbs measures given by the closed

convex hull of the set of weak limits:

Pβ,µ = limN→∞

Pβ,µN , (4.16)

and dene the domain of parameters where the weak limit Gibbs distribution exists

Γ =

(β, µ) ∈ R2+ : Gβ,µ 6= ∅

,

and domain where the weak limit Gibbs distribution exists and it is unique

Γ1 =

(β, µ) ∈ R2+ : |Gβ,µ| = 1

.

It is evident that Γ1 ⊆ Γ. Thus, the critical curve γcr for the Ising model coupled to CDT is

dened by

γcr = ∂Γ1 ∩ R2+. (4.17)

Let λ(β, µ) be given by

λ(β, µ) = c2 (m2 + 1) (cosh 2β)

(1 +

√1− 1

(cosh 2β)2

(m2 − 1)2

(m2 + 1)2

)(4.18)

where c and m are determined by

c =exp(β − µ)

e2β(1− exp(β − µ))2 − e−2µ(4.19)

m = e2β + (1− e4β) exp (−(β + µ)), (4.20)

Remember that identity (4.18) was derived in Chapter 3, Lemma 3.2.1.

We dene the strictly increasing function

ψ(β) = infµ ∈ R+ : λ(β, µ) < 1, for β > 0, (4.21)

34 FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT 4.3

and the following set

Σ =

(β, µ) ∈ R2

+ : µ < −3

2β + 2 ln 2

(β, µ) ∈ R2

+ : µ < −3

2β +

3

2ln(e2β − 1

)+ ln 2

.

Let t1, . . . , tk be triangulations of a single strip S× [0, 1] and σ1, . . . ,σk be their correspond-

ing spin congurations. Given 0 ≤ i1 < · · · < ik ≤ N − 1 we dene the nite-dimensional

cylinder Ci1,...,ik = C(t1,σ1),...,(tk,σk)i1,...,ik

as follows

Ci1,...,ik = (t,σ) : (t(i1),σ(i1)) = (t1,σ1), . . . , (t(ik),σ(ik)) = (tk,σk) (4.22)

Theorem 4.3.1. If (β, µ) ∈ Σ then there exists N0 ∈ N such that the partition func-

tion ΞN(β, µ) = +∞ whenever N > N0. Moreover, the Gibbs distribution Pβ,µN with periodic

boundary conditions cannot be dened by using the standard formula with ΞN(β, µ) as a nor-

malising denominator, consequently, there is no limiting probability measure Pβ,µ as N →∞.

Formally, for any nite-dimensional cylinder Ci1,...,ik we obtain Pβ,µN (Ci1,...,ik) = 0 whenever

N > N0 ≥ maxi1, . . . , ik.

Let β∗1 , β∗2 be positive solution of equations

− 3

2β + 2 ln 2 = −3

2β +

3

2ln(e2β − 1) + ln 2 (4.23)

and3

2β + 2 ln 2 = ψ(β), (4.24)

respectively. Together with results from before chapter (see [HYSZ13] for more details),

Theorem 4.3.1 provides two-side bounds for the critical curve.

Theorem 4.3.2. The critical curve γcr satises the following inequalities.

1. If (β, µ) ∈ γcr and 0 < β < β∗1 , then

−3

2β + 2 ln 2 ≤ µ < ψ(β).

The above bound implies that: For any sequence (βk, µk) ⊂ γcr such that βk → 0,

then limk→∞ µk = 2 ln 2.

2. If (β, µ) ∈ γcr and β∗1 ≤ β < β∗2 , then

3

2ln(e2β − 1)− 3

2β + ln 2 ≤ µ < ψ(β).

4.3 THE MAIN RESULTS 35

0 1 2 3 4 5 6 7 8

0

5

10

15

Figure 4.3: The area above the minimum of the dotted curve I (graph of the function ψ dened in

(4.21)) and dash-dotted line II is where the limiting Gibbs probability measure exists and is unique.

The critical curve lies in the region below the dotted curve I and dash-dotted line II but above the

continuous curve III and dashed line IV.

3. If (β, µ) ∈ γcr and β∗2 ≤ β <∞, then

3

2ln(e2β − 1)− 3

2β + ln 2 ≤ µ <

3

2β + 2 ln 2.

As a by-product of the proof of Theorems 4.3.1 and 4.3.2, using the FK representation

we also nd a lower and upper bound for the innite-volume free energy.

Corollary 4.3.1. If µ >3

2β+2 ln 2, then the free energy for the innite-volume Ising model

coupled to CDTs is nite and satises the following inequalities.

1. If 0 < β <1

3ln 2, then

ln Λ

(µ+

3

2β − ln 2

)≤ lim

N→∞

1

Nln ΞN(β, µ) ≤ ln Λ

(µ− 3

2β − ln 2

).

2. If1

3ln 2 ≤ β <∞, then

ln Λ

(µ− 3

)≤ lim

N→∞

1

Nln ΞN(β, µ) ≤ ln Λ

(µ− 3

2β − ln 2

).

Here Λ(s) is given by (2.10).

36 FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT 4.4

For each N ∈ N, we dene the follow set in R2+

ΓN = (β, µ) ∈ R2+ : KN is of trace class in `2

T−C, (4.25)

Γ− =⋂N∈N

ΓN and Γ+ =⋃N∈N

ΓN . (4.26)

Obviously, Γ− ⊂ ΓN ⊂ Γ+, for any N ≥ 1, and Pβ,µN there exist on ΓN . In order to each

N ≥ 1, we dene the N -strip functions fN associated with the partition function for the N

-strip Ising model coupled to CDTs as

fN(β) = infµ ∈ R2+ : (β, µ) ∈ ΓN for β > 0. (4.27)

According to Theorem 3.2.2 in before chapter , Theorem 4.3.2 and Proposition 4.4.4,

given in the Section 4.4.2, implies a similar version of Theorem 3.2.2, as following.

Theorem 4.3.3. For (β, µ) ∈ Γ+ = (β, µ) ∈ R2+ : µ > fT−C(β), the following limit holds:

limN→∞

1

Nln ΞN(β, µ) = ln Λ(β, µ), (4.28)

where Λ(β, µ) is the maximal eigenvalue of K and KT in `2T−C and fT−C is pointwise limit

of the family of functions fN. Consequently, as N → ∞ the N-strip Gibbs measure Pβ,µNconverges weakly to a limiting probability distribution Pβ,µ.

4.4 Proof of Theorem 4.3.1 and 4.3.2

The proof is based on nding of upper and lower bounds for the functions fN , introduced

in (4.27), using the FK representation (4.15) and the asymptotic behaviour of the partition

function ZN(·) for pure CDTs with periodical boundary condition. These bounds with the

Proposition 4.4.1, Proposition 4.4.2 and Proposition 4.4.3, established bounds for the critical

curve.

4.4.1 Proof of Theorem 4.3.1

We need two preparatory results. Let t be a Lorentzian CDT on cylinder CN . Given

1 ≤ i ≤ n(t), we dene the sets

Πi = all realization ξ of process ξ〈t,t′〉 such that k(ξ) = i. (4.29)

4.4 PROOF OF THEOREM ?? AND ?? 37

Thus, we have the following representation of (4.14)

Zβ,tN = e32βn(t)

n(t)∑i=1

2iPβ,t(Πi). (4.30)

Let ξ ∈ Πk and let Clkl=1 be the corresponding cluster decomposition of the set ∆(t).

Let ηl = η(Cl) and κl = κ(Cl) denote the number of vertices (triangles) in cluster Cl and the

number of edges in Cl, respectively. Note that κl depends on the geometry of cluster Cl.The probability that two nearest neighbor triangles t, t′ are linked is Pβ,t(t ↔ t′) =

1 − e−2β. Then, denoting p := 1 − e−2β, we obtain the following representation for the

probability of the set Πk,

Pβ,t(Πk) =∑

C1,...,Ck⊆∆(t)

p∑l κl (1− p)

32n(t)−

∑kl=1 κl

= (1− p)32n(t)

∑C1,...,Ck⊆∆(t)

(p

1− p

)∑kl=1 κl

.

(4.31)

Combining (4.31) with (4.30), we get the representation by cluster of the partition function

of Ising model supported by the triangulation t

Zβ,tN = e−32βn(t)

n(t)∑k=1

2k∑C1,...,Ck

(e2β − 1

)∑kl=1 κl . (4.32)

In order to obtain lower bounds for the critical curve, we employ the representation (4.32)

and consider several particular cases of interest.

The case k = n(t) : In this case there exists an unique way to decompose the set ∆(t)

in n(t) maximal connected components, considering clusters as isolated vertices Cl = t,t ∈ ∆(t), and 1 ≤ l ≤ n(t). This decomposition implies that κl = κ(Cl) = 0. Thus, by

relation (4.32), we obtain the following lower bounds for the partition function of the Ising

model on triangulation t

Zβ,tN ≥ e(− 32β+ln 2)n(t). (4.33)

Using (4.15), the lower bound in (4.33) provides the following lower bound to ΞN(β, µ),

ΞN(β, µ) ≥ ZN

(µ+

3

2β − ln 2

). (4.34)

Thus, using the asymptotic property given in Proposition (2.2.1) and Remark 2.2.2, we

38 FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT 4.4

obtain that the partition function ΞN(β, µ) there exists if

µ > −3

2β + 2 ln 2 + ln

(cos

π

N + 1

).

Letting N →∞ we obtain the following proposition.

Proposition 4.4.1. If (β, µ) ∈ R2+ such that µ < −3

2β + 2 ln 2 then there exists N0 ∈ N

such that the partition function ΞN(β, µ) = +∞ whenever N > N0. Moreover, the Gibbs

distribution Pβ,µN with periodic boundary conditions cannot be dened by using the standard

formula with ΞN(β, µ) as a normalising denominator, consequently, there is no limiting

probability measure Pβ,µ as N →∞. Futhermore, for any nite-dimensional cylinder Ci1,...,ikwe obtain Pβ,µN (Ci1,...,ik) = 0 whenever N > N0 ≥ maxi1, . . . , ik.

The case k = n(t) − 1 : This case is discussed here for an illustrative purpose. Notice

that in this case there exists 32n(t) ways to decompose the set ∆(t) in n(t) − 1 maximal

connected components: n(t) − 1 isolated vertices (triangles) and one cluster of two nearest

neighbor vertices (triangles). That is, if C is a cluster, then η(C) = 1 or 2. Moreover, for

each decomposition C1, . . . , Cn(t)−1 we have that∑n(t)−1

l=1 κl = 1 . This implies the following

inequality

Zβ,tN >1

2e(− 3

2β+ln 2)n(t)

∑C1,...,Cn(t)−1

(e2β − 1

)∑n(t)−1l=1 κl

=3

4

(e2β − 1

)n(t)e(− 3

2β+ln 2)n(t)

>3

4

(e2β − 1

)e(− 3

2β+ln 2)n(t), as n(t) ≥ 1.

(4.35)

Thus, we obtain another lower bound for the partition function of N -strip Ising model

coupled to CDTs

ΞN(β, µ) ≥ 3

4

(e2β − 1

)ZN

(µ+

3

2β − ln 2

). (4.36)

Therefore, in this case we get the same inequality that in Proposition 4.4.1.

It would be interesting to analyse a general case k = n(t)− l, but it seems that it won't

yield a better bound.

4.4 PROOF OF THEOREM ?? AND ?? 39

The case k = 1 : Consider the following subset of Π1:

Π(0)1 =

number of edges in cluster is κ1 =

3

2n(t)

∩Π1.

The probability of Π(0)1 is easy to calculate

Pβ,t(Π(0)1 ) =

(1− e−2β

) 32n(t)

.

Then, by relatio (4.32)

Zβ,tN > 2e−32βn(t)

(e2β − 1

) 32n(t)

= 2 exp

−(

3

2β − 3

2ln(e2β − 1

))n(t)

.

Thus

ΞN(β, µ) > 2∑

t e−µn(t) exp

−(

3

2β − 3

2ln(e2β − 1

))n(t)

= 2ZN

(µ+

3

2β − 3

2ln(e2β − 1

)).

(4.37)

As before, by asymptotic property (2.2.1), the partition function exists if

µ > −3

2β +

3

2ln(e2β − 1

)+ ln

(2 cos

π

N + 1

).

Letting N →∞ we obtain the following proposition.

Proposition 4.4.2. If (β, µ) ∈ R2+ such that µ < −3

2β +

3

2ln(e2β − 1) + ln 2 then there

exists N0 ∈ N such that the partition function ΞN(β, µ) = +∞ whenever N > N0. Moreover,

the Gibbs distribution Pβ,µN with periodic boundary conditions cannot be dened by using the

standard formula with ΞN(β, µ) as a normalising denominator, consequently, there is no

limiting probability measure Pβ,µ as N →∞. Futhermore, for any nite-dimensional cylinder

Ci1,...,ik we obtain Pβ,µN (Ci1,...,ik) = 0 whenever N > N0 ≥ maxi1, . . . , ik.

Proof of Theorem 4.3.1. The proof follows immediately from Proposition 4.4.1 and Propo-

sition 4.4.2.

40 FK REPRESENTATION FOR THE ISING MODEL COUPLED TO CDT 4.4

4.4.2 Proof of Theorem 4.3.2

The proof of Theorem 4.3.2 relies on two aditional obsevations. These are:

(1) Upper bounds for the functions fN and existence of the pointwise limit limN→∞ fN =

fT−C .

(2) The fact that graph of fT−C provides an upper bound for the critical curve.

Consequently, as by-product of Chapter 3 (see [HYSZ13]), we obtain the following assertions.

Proposition 4.4.3. For all N ∈ N, the following property of functions fN is fullled:

1. If 0 < β < β∗2 , then

fN(β) ≤ ψ(β), (4.38)

where β∗2 is positive solution of Eqn (4.24) and function ψ is introduced in Eqn (4.21).

2. If β∗2 ≤ β <∞, then

fN(β) ≤ 3

2β + 2 ln 2. (4.39)

Proposition 4.4.4. Functions fN converge pointwise:

fT−C(β) := limN→∞

fN(β) for β > 0. (4.40)

Combining (4.38), (4.39) with Proposition 4.4.4 and letting N → ∞, we obtain the

desired upper bound for the limit function fT−C

fT−C(β) ≤ ψ(β) if 0 < β < β∗2

fT−C(β) ≤ 3

2β + 2 ln 2 if β∗2 ≤ β <∞.

(4.41)

Since the graph of fT−C lies above the critical curve, the right-hand side of (4.41) provides

an upper bound for the critical curve.

Proof of Theorem 4.3.2. The upper bound of Theorem 4.3.2 is consequence of Eqn (4.41).

The lower bound is consequence of Proposition 4.4.1 and 4.4.2. This concludes the proof of

Theorem 4.3.2.

Chapter 5

Potts model coupled to CDTs and FK

representation

In this chapter using a natural generalization of Ising model, we extend results from

before chapters for the (annealed) classical Ising model coupled to two-dimensional causal

dynamical triangulations. Such generalization is called of Potts model. Whereas in Ising sys-

tems the spins on two dierent values, in the q-state Potts model q distinct values, represent

by the elements of the set 1, . . . , q, are allowed on any vertex from the triangulation t.

In Chapter 3 and 4, the Ising model was dened putting spins on any triangles (faces), but

it is equivalent to put spins on any vertex of dual triangulation, dened in Section 5.2.2. Using

duality relation on a torus (periodic boundary condition), we provide a relation between the

free energy of Potts model coupled to CDTs and Potts model coupled to DUAL CDTs.

Additionally, using the high temperature expansion (and duality relation), we determined a

region where the critical curve can be located. This bound serves for Ising model case, and

improves the bounds found in the before chapters (Chapter 3: Theorem 4.3.1 and Theorem

4.3.2. Chapter 4: Lemma 3.2.1 and Theorem 3.2.2).

5.1 Introduction and main results of this chapter

A causal dynamical triangulation (CDT), introduced by Ambjørn and Loll (see [AL98]),

together with its predecessor a dynamical triangulation (DT), constitute attemps to provide

a meaning to formal expressions appearing in the path integral quantisation of gravity (see

[ADJ97], [AJ06] for an overview). The idea is to regularise the path integral by approximat-

ing the geometries emerging in the integration by CDTs. As a result, the path integral over

geometries is replaced with a sum over all possible triangulations where each conguration

is weighted by a Boltzmann factor e−µ|T |, with |T | standing for the size of the triangula-

tion and µ being the cosmological constant. The evaluation of the partition function was

reduced to a purely combinatorial problem that can be solved with the help of the early

work of Tutte [Tut62, Tut63]; alternatively, more powerful techniques were proposed, based

41

42 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.1

on random matrix models (see, e.g., [FGZJ95]) and bijections to well-labelled trees (see

[Sch97, BDG02]).

From a physical point of view it is interesting to study various models of matter, such as

the q-state Potts model, coupled to the CDT. The goal of this chapter is dene the q-state

Potts model coupled to CDTs and will use the FK representation for study this model. In this

case, the calculation of the partition function also reduces to a combinatorial problem. For

the 2-state Potts model (Ising model) coupled to a CDT some progress has been recently

made on existence of Gibbs measures and phase transitions (see [AAL99], [BL07], [HYSZ13]

and [Her14] for details). In particular, using transfer matrix methods, the Krein-Rutman

theory and FK representation for the Ising model, [Her14] provides a region in the quadrant

of parameters β, µ > 0 where the innite-volume free energy has a limit, providing results

on convergence and asymptotic properties of the partition function and the Gibbs measure.

Thus, FK-Potts models, introduced by Fortuin and Kasteleyn (see [FK72]), prove that these

models have become an important tool in the study of phase transition for the Ising and

q-state Potts model.

In general, the FK-Potts model on a nite connected graph (not necessarily planar) is

a model of edges of the graphs, each edge is either closed or open. The probability of a

conguration is proportional to

p#open edges(1− p)#closed edgesq#clusters

,

where the edge-weight p ∈ [0, 1] and the cluster-weight q ∈ (0,∞) are the parameters of the

model. For q ≥ 1, this model can be extended to innite-volume where it exhibits a phase

transition at some critical parameter pc(q), that depend on the geometry of the graph. In the

case of planar graphs, there is a connection between FK-Potts models on a graph and on its

dual with the same cluster-weight q and appropriately related edge-weight p and p∗ = p∗(p)

(Kramers-Wannier duality). For example, this relation leads in the particular case of Z2 to

a natural conjecture: the critical point is the same as the so-called self-dual point satisfying

psd = p∗(psd), proved by Beara and Duminil-Copin in [BC12].

In the case of a FK-Potts model dened on a causal dynamical triangulation t with pe-

riodic boundary condition, or equivalently dened on a torus (see Figure 3.1 for a geometric

representation), its dual, dened on t∗, is not a FK-Potts model; but will enough for our

purposes. This relation together with the Edwars-Sokal coupling, using p = 1− e−β, permits

nd a relation between the parameters (β, µ) of the Potts model coupled to CDT and the

parameters (β∗, µ∗) of its dual for the innite-volume (thermodynamic limit).

In the present chapter, we prove the following duality relation.

Theorem 5.1.1. Let q ≥ 2. The free energy of the q-state Potts model coupled to causal

5.1 INTRODUCTION AND MAIN RESULTS OF THIS CHAPTER 43

(a) (b)

Figure 5.1: Illustrating the region where the critical curve for Potts model coupled CDTs and dual

CDTs can be located.

dynamical triangulation and its dual satised the following duality relation

limN→∞

1

Nln ΞN(β, µ) = lim

N→∞

1

Nln Ξ∗N(β∗, µ∗) (5.1)

where ΞN , Ξ∗N denote the partition function of the q-state Potts model coupled to CDT and

coupled dual CDT respectively (dened in Section 5.2.1), and

β∗ = ln

(1 +

q

eβ − 1

), µ∗ = µ− 3

2ln(eβ − 1) + ln q. (5.2)

Thus, (5.1) relates the free energy of the q-state Potts model coupled to CDTs and the

free energy of the q-state Potts model coupled to dual CDTs, and maps the high and low

temperature of the dual models onto each other.

We will use the duality relation of Theorem 5.1.1 and the high-temperature expansion

for the q-state Potts model for determine a region in the quadrant of parameters where the

critical curve for the q-state Potts model coupled CDTs and q-state Potts model coupled

dual CDTs can be located (see Figure 5.1).

Understanding by critical curve of the model the boundary of the domain of parameters

β and µ (β∗ and µ∗ on its dual, respectively) where the model exhibits subcritical behavior

(see denition in Section 4.3 of Chapter 4), this chapter makes a rigorous derivation of the

subcriticality domain for an q-Potts model coupled to two-dimensional CDT and a domain

where the tipical innite-volume Gibbs measure there no exists. The proof involve two tech-

niques: the duality relation (Theorem 5.1.1) and high-temperature expansion for the q-state

Potts model. In Figure 5.1, we show the region where the critical curve of the model should

be located (gray region).

44 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.1

Dene the sets

Σ =

(β, µ) ∈ R2

+ : µ < max

ln(2√q),

3

2ln(eβ − 1

)+ ln 2

,

and

Σ∗ =

(β∗, µ∗) ∈ R2

+ : µ∗ < max

ln(2q),

3

2ln(eβ∗ − 1

)+ ln 2

.

We prove the following long theorem for existence and no existence of Gibbs measure for

the model.

Theorem 5.1.2. Let q ≥ 2.

1. Potts model coupled to CDTs. If (β, µ) ∈ Σ then there exists N0 ∈ N such that the par-

tition function ΞN(β, µ) = +∞ whenever N > N0. Moreover, the Gibbs distribution

Pβ,µN with periodic boundary conditions cannot be dened by using the standard for-

mula with ΞN(β, µ) as a normalising denominator, consequently, there is no limiting

probability measure Pβ,µ as N →∞. Furthermore, if (β, µ) satised

µ >3

2ln(q + eβ − 1

)+ ln 2− ln q +

3

2ln

(1 + (q2/3 − 1)

eβ − 1

q + eβ − 1

), (5.3)

the innite-volume free energy exists, i.e. the following limit there exists:

limN→∞

1

Nln ΞN(β, µ).

Moreover, as N →∞, the Gibbs distribution Pβ,µN converges weakly to a limiting prob-

ability distribution Pβ,µ.

2. Potts model coupled to dual CDTs. If (β∗, µ∗) ∈ Σ∗ then we have the same conclusion

for the the Gibbs distribution Pβ∗,µ∗

N , i.e. there is no limiting probability measure Pβ∗,µ∗

as N →∞. Furthermore, if (β∗, µ∗) satised

µ∗ >3

2β∗ + ln 2 +

3

2ln

(1 +

q2/3 − 1

eβ∗

), (5.4)

the innite-volume free energy exists and, as N → ∞, the Gibbs distribution Pβ∗,µ∗

N

converges weakly to a limiting probability distribution Pβ∗,µ∗.

As a byproduct, the Theorem 5.1.2 serves to nd lower and upper bounds for the innite-

volume free energy. Moreover, in the case of 2-state Potts model (Ising model), Theorem

5.1.2 extends earlier results from [Her14], [HYSZ13] and improves the approximation of the

curve in high temperature given in [AAL99]. In aditional, this approach allows to get a

better aproximation of the critical curve and check the asymptotic behavior of the critical

5.2 NOTATIONS 45

curve given in [AAL99], and it say that critical curve is asymptotic to 32β + ln 2, for q ≥ 2.

In Theorem 5.1.2, we nd a lower and upper curve that converges fast to 32β + ln 2.

5.2 Notations

In this section we rts introduce notations and give a summary of q-state Potts model and

we dene the Potts model coupled to CDTs. Finally, we give a short review of the Edwards-

Sokal coupling. We refer to [MYZ01], [Gri06], [HYSZ13], for more details. We attempt at

establishing regions where the innite-volume free energy converges, yielding results on the

convergence and asymptotic properties of the partition function and the Gibbs measure.

5.2.1 A Potts model coupled to CDTs

Let t be a CDT on the cylinder CN with periodic boundary condition. Each triangulation

t can be view as a graph t = (V (t), E(t)) embedded on a torus. Potts spin systems are

generalizations of the Ising model. Whereas in Ising systems the spins on two dierent

values, in the q-state Potts model q distinct values, represent by the elements of the set

1, . . . , q, are allowed on any vertex from the triangulation t. We consider the product

sample space Ω(t) = 1, . . . , qV (t) and we consider a usual (ferromagnetic) q-state Potts

model energy, where two spins σ(t) and σ(t′) interact if their supporting vertices t, t′ are

connected by an common edge; such vertices are called nearest neighbors, and this property

is reected in the notation 〈t, t′〉. Thus, the Hamiltonian used for the q-state Potts model

on t is given by

h(σ) = −∑〈t,t′〉

δσ(t),σ(t′). (5.5)

The partition function for the q-state Potts model on t is dene by

ZP (β, q, t) =∑σ

exp−βh(σ)

, (5.6)

where the summation is over any congurations σ ∈ 1, . . . , qV (t). Thus, the q-state Potts

measure on t is dene as follows

µtβ,q(σ) =

1

ZP (β, q, t)exp−βh(σ)

. (5.7)

Using the partition function for the q-state Potts model on a xed t, we dene the partition

function for the q-state Potts model coupled to CDTs, at the inverse temperature β > 0 and

the cosmological constant µ, as follows

ΞN(β, µ) =∑t

exp−µn(t)

ZP (β, q, t) (5.8)

46 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.2

where n(t) stands for the number of triangles in the triangulation t. Similarly, we introduce

the N -strip Gibbs probability distribution associated with (5.8)

Pβ,µN (t,σ) =1

ΞN(β, µ)exp−µn(t)− βh(σ)

. (5.9)

and we denote by Gβ,µ the set of Gibbs measures given by the closed convex hull of the set

of weak limits:

Pβ,µ = limN→∞

Pβ,µN , (5.10)

In general, the q-state Potts model can be dened on a general lattice G. Therefore, it

is possible dene the q-state Potts model sobre the dual t∗ of the triangulation t (see next

section for a formal denition of t∗). The partition function for the q-state Potts model on

t∗ will denote by ZP (β∗, q, t∗). Finally, we dene the partition function for the q-state Potts

model coupled to dual CDTs, Ξ∗N(β∗, µ∗) as follow

Ξ∗N(β∗, µ∗) =∑t

exp−µ∗n(t)

ZP (β∗, q, t∗). (5.11)

5.2.2 The FK-Potts model on Lorentzian triangulations

We now turn to the FK representation of the q-state Potts model. The random cluster

model was originally introduced by Fortuin and Kasteleyn [FK72] and it can be understood

as an alternative representation of the q-state Potts model. This representation will be

referred to as the FK representation or FK-Potts model. We are interested in study FK-

Potts model on CDTs and dual CDTs, and nd a duality relation relation between the

parameters of the model on CDTs and its dual. In [HYSZ13], [Her14], the model was dened

putting spins on any triangle (faces), but it is equivalent to put spins on any vertex of dual

graph, in this case, dual triangulation. In this section we work with triangulations with

periodic boundary conditions, i.e., Lorentzian triangulations embedded in a torus T (see

Figure 3.1 (b)) and its dual. In general, let G = (V,E) be a graph embedded in T, we obtainits dual graph G∗ = (V ∗, E∗) as follows: we place a dual vertex within each face of G. For

each e ∈ E we place a dual e∗ = 〈x∗, y∗〉 joining the two dual vertices lying in the two faces

of G abutting e. Thus, V ∗ is in one-one correspondence with the set of faces of G, and E∗ is

a one-one correspondence with E. For each Lorentzian triangulation t, we denote by t∗ its

dual.

Let t = (V (t), E(t)) be a Lorentzian triangulation with periodic boundary condition,

where V (t), E(t) denote the set of vertices and edges, respectively. The state space for the

FK-Potts model is the set Σ(t) = 0, 1E(t), containing congurations that allocate 0′s and

1′s to the edge e = 〈i, j〉 ∈ E(t). For w ∈ Σ(t), we call an edge e open if w(e) = 1, and closed

if w(e) = 0. For w ∈ Σ(t), let η(w) = e ∈ E(t) : w(e) = 1 denote the set of open edges.

Thus, each w ∈ Σ(t) splits V (t) into the disjoint union of maximal connected components,

which are called the open clusters of Σ(t). We denote by k(w) the number of connected

5.2 NOTATIONS 47

Figure 5.2: Geometric representation of a dual Lorentzian triangulation t∗ with periodic spatial

boundary condition.

components (open clusters) of the graph (V (t), η(w)), and note that k(w) includes a count

of isolated vertices. Two sites of t are said to be connected if one can be reached from another

via a chain of open bonds.

The partition function of the FK-Potts model on t with parameters p and q and periodic

boundary condition is dened by

ZFK(p, q, t) =∑

w∈Σ(t)

∏e∈E(t)

(1− p)1−w(e)pw(e)

qk(w), (5.12)

Thus, the FK-Potts measure on t is dene as follows

Φtp,q(w) =

1

ZFK(p, q, t)

∏e∈E(t)

(1− p)1−w(e)pw(e)

qk(w). (5.13)

We will use a similarly notation for the FK-Potts model on dual triangulation t∗. We

denote by ZFK(p∗, q, t∗) and Φt∗p∗,q the partition function and the FK-Potts measure on t∗

with parameters p∗ and q, respectively.

5.2.3 The relation between the Potts model and FK-Potts model:

Edwards-Sokal coupling

There are several ways to make the connection between the Potts and FK-Potts model.

The correspondence between the q-state Potts model and FK-Potts model was established

by Fortuin and Kasteleyn [FK72] (see also [ES88], [Gri06]). In a modern approach, these two

models are related via a coupling, i.e., coupled the two systems on a common probability

space. This coupling was introduced by Edwards-Sokal in [ES88].

Let t be a CDT on the cylinder CN with periodic boundary condition. We consider the

48 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.2

product sample space Ω(t)×Σ(t) where Ω(t) = 1, 2, . . . , qV (t) and Σ(t) = 0, 1E(t). The

Edwards-Sokal measure Q on Ω(t)× Σ(t) is dene by

Q(σ,w) ∝∏

e=i,j∈E(t)

(1− p)δw(e),0 + pδw(e),1δσi,σj

Theorem 5.2.1 (Edwards-Sokal [ES88]). Let q ∈ 2, 3, . . . . Let p ∈ (0, 1) and t a CDT with

periodic boundary condition, and suppose that p = 1−e−β. If the conguration w is distributed

according to an FK-Potts measure with parameters (p, q) on t, then σ is distributed according

to a q-state Potts measure with inverse temperature β. Furthermore, the Edwards-Sokal

measure provides a coupling of µtβ,q and Φt

p,q, i.e.∑w∈Σ(t)

Q(σ,w) = µtβ,q(σ),

for all σ ∈ Ω(t), and ∑σ∈Ω(t)

Q(σ,w) = Φtp,q(w),

for all w ∈ Σ(t). Moreover, we have the relation between partition functions

ZFK(p, q, t) = e−β|E(t)|ZP (β, q, t). (5.14)

5.2.4 Duality for FK-Potts model coupled to CDTs with periodic

boundary conditions

In this section we obtain a relation between the partition functions of FK-Potts model

on a triangulation t and its dual. This relation was studied by Beara and Duminil-Copin

for the FK-Potts model on Z2 with free, wired and periodic boundary condition (see [BC12]

for details). We will view wich the dual of a FK-Potts model dened on a torus is a quasi

FK-Potts model, but it is not very dierent from one.

Let t and t∗ a CDT with periodic boundary condition and its dual. Each conguration

w ∈ Σ(t) = 0, 1E(t) gives rise to a dual conguration w∗ ∈ Σ(t∗) = 0, 1E(t∗) given by

w∗(e∗) = 1 − w(e). That is, e∗ is declared open if and only if the corresponding bond e is

closed. The new conguration w∗ is called the dual conguration of w, and note that there

exists an one-one correspondence between Σ(t) and Σ(t∗). As in the Section 5.2.2, to each

conguration w∗ there corresponds the set η(w∗) = e∗ ∈ E(t∗) : w∗(e∗) = 1 of its openedges.

Now, beginning of FK-Potts model on t, we try to obtain the dual model on the dual

triangulation t∗.

Let o(w) (resp. c(w)) denote the number of open edges (resp. closed) of w, k(w) the

5.2 NOTATIONS 49

(a) (b) (c)

Figure 5.3: (a) Geometric representation of a net (b) Geometric representation of a cycle (c) None

of cluster of w is a net or a cycle

number of connected components of w, and f(w) the number of faces delimited by w, i.e.

the number of connected components of the complement of the set of open bonds. We will

now dene an additional parameters δ(w).

Call a connected component of w a net if it contains two non-contractible simple loops

γ1, γ2 of dierent homotopy classes, and a cycle if it contain a non-contractible simple loops

γ1 non-contractible but is not a net (see Figure 5.3). These denitions were introduced in

[BC12]. In aditional, notice that every conguration w can be of one three types:

• One of the cluster of w is a net. Then no other cluster can be a net or a cycle. In that

case, we let δ(w) = 2;

• One of the cluster of w is a cycle. Then no other cluster can be a net, but other cluster

can be cycles as well (in which case all the involved, simple loops are in the same

homotopy class) We then let δ(w) = 1;

• None of the cluster of w is a net or a cycle. We let δ(w) = 0.

Using this denition for the parameter δ, we obtained the following version of Euler's

formula.

Proposition 5.2.1 (Euler's formula). Let t a CDT with periodic boundary condition and

w ∈ 0, 1E(t). Then

|V (t)| − o(w) + f(w) = k(w) + 1− δ(w). (5.15)

Using duality and Proposition 5.2.1, we have the following relations

o(w) + o(w∗) = |E(t)|, f(w) = k(w∗) and δ(w) + δ(w∗) = 2. (5.16)

50 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.2

Let q ∈ (0,∞) and p ∈ (0, 1). The partition function of the FK-Potts model is given by

ZFK(p, q, t) =∑

w∈Σ(t)

∏e∈E(t)

(1− p)1−w(e)pw(e)

qk(w)

=∑

w∈Σ(t)

po(w)(1− p)c(w)qk(w).

Using Euler's formula and relations (5.16), we rewrite the number of cluster of w in terms

of its dual w∗

k(w) = |V (t)| − |E(t)|+ o(w∗) + k(w∗) + 1− δ(w∗).

We note also that o(w) + o(w∗) = |E(t)| = |E(t∗)|. Plugging before relations into the

partition function of the FK-Potts model, we obtain

ZFK(p, q, t) =∑

w∈Σ(t)

po(w)(1− p)|E(t)|−o(w)qk(w)

= (1− p)|E(t)|∑

w∈Σ(t)

(p

1− p

)o(w)

qk(w)

= (1− p)|E(t)|∑

w∈Σ(t)

(p

1− p

)|E(t)|−o(w∗)

q|V (t)|−|E(t)|+o(w∗)+k(w∗)+1−δ(w∗)

= p|E(t)|q|V (t)|−|E(t)|∑

w∈Σ(t)

(1− pp

)o(w∗)qo(w

∗)+k(w∗)+1−δ(w∗)

As there exists an one-one correspondence between Σ(t) and Σ(t∗), in the last equality, we

we change the sum in Σ(t) by the sum in Σ(t∗). Thus, we obtain the following representation

of the partition function in terms of the dual triangulation and dual congurations

ZFK(p, q, t) = p|E(t)|q|V (t)|−|E(t)|∑

w∗∈Σ(t∗)

(q(1− p)

p

)o(w∗)qk(w∗)+1−δ(w∗) (5.17)

Using the relation (5.17), we obtain the following lemma.

Lemma 5.2.1. Let t be a CDT with periodic boundary condition. Then the following com-

parison inequalities both

ZFK(p, q, t) ≤(

p

1− p∗

)|E(t)|

q|V (t)|−|E(t)|+1ZFK(p∗, q, t∗) (5.18)

5.2 NOTATIONS 51

and (p

1− p∗

)|E(t)|

q|V (t)|−|E(t)|−1ZFK(p∗, q, t∗) ≤ ZFK(p, q, t) (5.19)

where ZFK(p∗, q, t∗) is the partition function for FK-Potts model on t∗ with parameters q

and p∗ = p∗(p, q) satisfying

p∗(p, q) =(1− p)q

(1− p)q + p, or equivalently

p∗

1− p∗· p

1− p= q.

Proof. We introduce the parameter p∗ = p∗(p, q) as solution of the equation

p∗

1− p∗=

(1− p)qp

.

Thus, the partition function can be written in the following ways

ZFK(p, q, t) = p|E(t)|q|V (t)|−|E(t)|∑

w∗∈Σ(t∗)

(p∗

1− p∗

)o(w∗)qk(w∗)+1−δ(w∗)

=p|E(t)|

(1− p∗)|E(t∗)| q|V (t)|−|E(t)|(1− p∗)|E(t∗)|

∑w∗∈Σ(t∗)

(p∗

1− p∗

)o(w∗)qk(w∗)+1−δ(w∗).

Notice that −1 ≤ 1 − δ(w∗) ≤ 1, for all w∗ ∈ Σ(t∗). We dene ZFK(p∗, q, t∗), the partition

function of a FK-Potts model with parameters p∗ and q. Thus, we obtain the upper bound

ZFK(p, q, t) ≤ p|E(t)|

(1− p∗)|E(t∗)| q|V (t)|−|E(t)|+1ZFK(p∗, q, t∗) ,

and the lower bound

p|E(t)|

(1− p∗)|E(t∗)| q|V (t)|−|E(t)|−1ZFK(p∗, q, t∗) ≤ ZFK(p, q, t)

for the partition function of FK-Potts model on t with parameters p and q. Using the one-one

correspondence between E(t) and E(t∗), we conclude the proof.

The partition function for pure CDT's has been determined as a sum over all possible

triangulations of a cylinder where each conguration is weighted by a Boltzmann factor

e−µn(t), where n(t) standing for the size of the triangulation and µ being the cosmological

constant. Thus, in quantum gravity the volume n(t) becomes a dynamical variable for the

model. Therefore, we rewrite the duality relation ( Lemma 5.2.1) for the partition function

of the FK-Potts model on a triangulation t, in terms of dynamical variable n(t). In the Table

5.1 we show the relation among V (t), E(t), V (t∗), E(t∗) and the number of triangles n(t) of

a CDT t.

52 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.3

t = (V (t), E(t)) t∗ = (V (t∗), E(t∗))

|V (t)| = 1

2n(t) |V (t∗)| = n(t)

|E(t)| = 3

2n(t) |E(t∗)| = 3

2n(t)

|faces in t| = n(t) |faces in t∗| = 1

2n(t)

Table 5.1: Relation between the graphs t, t∗ and n(t)

Using the relations of Table 5.1, the Lemma 5.2.1 becomes be written in terms of n(t)

as follow

Corollary 5.2.1. Let t be a CDT with periodic boundary condition. Then the following

comparison inequalities both

(p

1− p∗

) 32n(t)

q−1−n(t) ≤ ZFK(p, q, t)

ZFK(p∗, q, t∗)≤(

p

1− p∗

) 32n(t)

q1−n(t) (5.20)

and (p∗

1− p

) 32n(t)

q−1− 12n(t) ≤ ZFK(p∗, q, t∗)

ZFK(p, q, t)≤(

p∗

1− p

) 32n(t)

q1− 12n(t) (5.21)

where ZFK(p∗, q, t∗) is the partition function for FK-Potts model on t∗ with parameters q

and p∗ = p∗(p, q) satisfying

p∗(p, q) =(1− p)q

(1− p)q + p, or equivalently

p∗

1− p∗p

1− p= q.

5.3 The proof of Theorem 5.1.1 and rst bounds for the

critical curve

In the previous section we found comparison inequalities between the partition function

of the FK-Potts model on t and the partition function of the FK-Potts model on its dual t∗.

In this section we will use these comparison inequalities to prove Theorem 5.1.1. Combining

inequalities (5.20), (5.21) and the Edwars Sokal coupling (Theorem 5.2.1), we obtain the

following comparison inequalities between the partition function of the q-state Potts model

on t and the partition function of the q-state Potts model on its dual t∗.

(p

1− p∗

) 32n(t)

q−1−n(t)e32

(β−β∗)n(t) ≤ ZP (β, q, t)

ZP (β∗, q, t∗)≤(

p

1− p∗

) 32n(t)

q1−n(t)e32

(β−β∗)n(t)

(5.22)

5.3 THE PROOF OF THEOREM 5.1.1 AND FIRST BOUNDS FOR THE CRITICAL CURVE 53

and(p∗

1− p

) 32n(t)

q−1− 12n(t)e

32

(β∗−β)n(t) ≤ ZP (β∗, q, t∗)

ZP (β, q, t)≤(

p∗

1− p

) 32n(t)

q1− 12n(t)e

32

(β∗−β)n(t)

(5.23)

where (eβ − 1)(eβ∗ − 1) = q.

Proof of Theorem 5.1.1. Using the comparison inequalities (5.22) and (5.23), we will nd

comparison inequalities for the partition functions of the Potts model coupled CDTs with

parameters β, µ and dual CDTs with parameters β∗ = β∗(β), µ∗ = µ∗(β, µ). Remember that

p∗ = 1− e−β∗ and p = 1− e−β. Thus,

p∗

1− p= (1− e−β∗) + qe−β

∗=

q

(1− e−β) + qe−β

andp

1− p∗=

q

(1− e−β∗) + qe−β∗= (1− e−β) + qe−β.

Multiplying by the Boltzmann factor e−µn(t) in (5.22) and (5.23), and sum over all possible

CDTs of the cylinder CN , we obtain the following comparison inequalities

1

qΞ∗N(β∗, µ∗) ≤ ΞN(β, µ) ≤ qΞ∗N(β∗, µ∗) (5.24)

where Ξ∗N stands the partition function of the q-state Potts model coupled to dual CDTs

with periodic boundary condition, ΞN stands the partition function of the q-state Potts

model coupled to CDTs with periodic boundary condition, and

β∗ = ln

(1 +

q

eβ − 1

), µ∗ = µ− 3

2ln(eβ − 1) + ln q.

Similarly, we have1

qΞN(β, µ) ≤ Ξ∗N(β∗, µ∗) ≤ qΞN(β, µ) (5.25)

where

β = ln

(1 +

q

eβ∗ − 1

), µ = µ∗ − 3

2ln(eβ

∗ − 1) +1

2ln q.

Take the natural logarithm in inequalities (5.24) and (5.25), divide both sides of the above

inequalities by N and let N →∞. This concludes the proof of Theorem 5.1.1.

Theorem 5.1.1 provide an interesting reformulation in terms of free energy of the q-state

Potts model coupled to CDTs and its dual. This theorem relates the free energy of the q-

state Potts model coupled CDTs and the free energy of the q-state Potts model coupled dual

CDTs, and maps the high and low temperature of the dual models onto each other.

54 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.3

Using Edwars-Sokal coupling for the partition functions (5.14), duality relation found in

Theorem 5.1.1 and asymptotic properties (2.14), (3.20) of the partition function for pure

CDTs (see [MYZ01] for more details), we will obtain the rst bounds for the critical curve

of the q-state Potts model coupled to CDTs and dual CDTs. Let t be a CDT with periodic

boundary condition. We dene the set Πi of congurations in Σ(t) which splits V (t) in i

maximal connected components, i.e.

Πi = w ∈ Σ(t) : k(w) = i.

Similarly, we denote Π∗i the set of congurations in Σ(t∗) which splits V (t∗) in i maximal

connected components. Thus, we have the following representation for the partition function

of q-state Potts model on t

ZP (β, q, t) = eβ|E(t)|φtp(q

k(w)) = eβ|E(t)||V (t)|∑i=1

qiφtp(Πi), (5.26)

and on t∗

ZP (β∗, q, t∗) = eβ|E(t∗)|φt∗

p (qk(w∗)) = eβ|E(t∗)||V (t∗)|∑i=1

qiφt∗

p∗(Π∗i ), (5.27)

where φtp, φ

t∗p∗ denotes product measures on Σ(t) and Σ(t∗), respectively.

Using Table 5.1, we write the representations (5.26), (5.27) for the partition function in

terms of the dynamical variable n(t). For that, we consider two cases of interest separately.

1. The model on CDTs t: In this case, we can to write the partition function in terms of

volume n(t) of the triangulation as follow

ZP (β, q, t) = e32βn(t)

12n(t)∑i=1

qiφtp(Πi). (5.28)

Using the rst and latter term on the right-hand side of (5.28), we obtain two lower bounds

for the partition function of the Potts model on t

ZP (β, q, t) ≥ q(eβ − 1

) 32n(t)

, ZP (β, q, t) ≥ q12n(t). (5.29)

These lower bounds for the q-state Potts model on t permit to obtain a lower barrier for

parameters where the model can be dened, and the partition function of the model coupled

to CDTs could no explode in nite volume. These lower bounds serves to obtain information

of the Gibbs measure for q-state Potts model coupled to CDTs.

5.3 THE PROOF OF THEOREM 5.1.1 AND FIRST BOUNDS FOR THE CRITICAL CURVE 55

Proposition 5.3.1. If (β, µ) ∈ R+ such that

µ <1

2ln q + ln 2 or µ <

3

2ln(eβ − 1) + ln 2,

then there exists N0 ∈ N such that the partition function ΞN(β, µ) = ∞ whenever N >

N0. Moreover, the Gibbs distribution Pβ,µN cannot be dened by using the standard formula

with ΞN(β, µ) as a normalising denominator, consequently, there is no limiting probability

measure Pβ,µ as N →∞.

Proof. The lower bounds in (5.29) to ZP (β, q, t) provide the following lower bounds to

ΞN(β, µ),

ΞN(β, µ) ≥ q∑t

e−µ−32

ln(eβ−1)n(t), ΞN(β, µ) ≥ q∑t

e−µ−12

ln qn(t). (5.30)

Using asymptotic properties of Proposition (2.2.1), we obtain which the partition function

ΞN(β, µ) there is no exist if

µ ≤ 1

2ln q + ln

(2 cos

π

N + 1

)or µ ≤ 3

2ln(eβ − 1) + ln

(2 cos

π

N + 1

).

Letting N →∞, we conclude the proof.

Now, notice that12n(t)∑i=1

qiφtp(Πi) ≤ q

12n(t),

for any triangulation t. Thus, we obtain a upper bound for the partition function of the

q-state Potts model on t,

ZP (β, q, t) ≤ e32βn(t)q

12n(t) = e(

32β+ 1

2ln q)n(t). (5.31)

This upper bound for the q-state Potts model on t, permit to obtain a rst upper barrier

for the critical curve of the q-state Potts model coupled to CDTs, and above of that upper

bound the model exhibits subcritical behavior. Moreover, this upper bound for the critical

curve of the model coupled serves to obtain information of the Gibbs measure for the q-state

Potts model coupled to CDTs. We get the following result.

Proposition 5.3.2. Under condition µ >3

2β +

1

2ln q+ ln 2, the innite-volume free energy

exists, i.e. the following limit there exists:

limN→∞

1

Nln ΞN(β, µ).

Moreover, as N →∞, the Gibbs distribution Pβ,µN converges weakly to a limiting probability

distribution Pβ,µ.

56 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.3

Proof. Using inequality (5.31), we get to

ΞN(β, µ) ≤∑t

e−µ−32β− 1

2ln qn(t)

By Proposition 2.2.1, we obtain which the free energy there exists if µ− 32β − 1

2ln q > ln 2.

This concludes the proof.

Finally, using the duality relation (Theorem 5.1.1) and bounds found before propositions,

we obtain bounds for the critical curve for the q-state Potts model coupled to dual CDTs.

from CDTs tby duality−−−−−−−−→ to dual CDTs t∗

µ <1

2ln q + ln 2 → µ∗ <

3

2ln(eβ

∗− 1) + ln 2

µ <3

2ln(eβ − 1) + ln 2 → µ∗ < ln q + ln 2

µ >3

2β +

1

2ln q + ln 2 → µ∗ >

3

2ln(q + eβ

∗− 1) + ln 2

Table 5.2: Bounds for the critical curve of the q-state Potts model on CDTs will generate bounds

on its dual.

In Table 5.2 the parameters (β, µ) and (β∗, µ∗) satised the duality relation (5.2).

2. The model on dual CDTs t∗: Similarly, we write the partition function in terms of

volume n(t) of the triangulation as follow

ZP (β∗, q, t∗) = e32β∗n(t)

n(t)∑i=1

qiφt∗

p∗(Π∗i ). (5.32)

Using the rst and latter term on the right-hand side of (5.32), we obtain two lower bounds

for the partition function of the q-state Potts model on t∗

ZP (β∗, q, t∗) ≥ q(eβ∗ − 1

) 32n(t)

, ZP (β∗, q, t∗) ≥ qn(t), (5.33)

and an upper bound

ZP (β∗, q, t∗) ≤ e32β∗n(t)qn(t) = e(

32β∗+ln q)n(t). (5.34)

Using asymptotic properties of Theorem 2.2.1 and Proposition 2.2.1, bounds (5.33) and

(5.34) provide bounds for the critical curve of the q-state Potts model coupled dual CDTs.

As in before case, we have the following proposition to existence and non existence of Gibbs

measures of the model.

5.4 THE PROOF OF THEOREM 5.1.1 AND FIRST BOUNDS FOR THE CRITICAL CURVE 57

Proposition 5.3.3. For q-state Potts model coupled to dual CDTs, we have the following

assertions:

1. If (β∗, µ∗) ∈ R+ such that

µ∗ < ln q + ln 2 or µ∗ <3

2ln(eβ

∗ − 1) + ln 2,

then there exists N0 ∈ N such that the partition function Ξ∗N(β∗, µ∗) = ∞ whenever

N > N0. Moreover, the Gibbs distribution Pβ∗,µ∗

N cannot be dened by using the stan-

dard formula with Ξ∗N(β∗, µ∗) as a normalising denominator, consequently, there is no

limiting probability measure Pβ∗,µ∗ as N →∞.

2. Under condition µ∗ >3

2β∗ + ln q + ln 2, the innite-volume free energy exists, i.e. the

following limit there exists:

limN→∞

1

Nln Ξ∗N(β∗, µ∗).

Moreover, as N → ∞, the Gibbs distribution Pβ∗,µ∗

N converges weakly to a limiting

probability distribution Pβ∗,µ∗.

Finally, using the duality relation (Theorem 5.1.1), and bounds found in the before propo-

sition, we obtain bounds for the critical curve for the Potts model coupled to CDTs. In Table

from dual CDTs t∗by duality−−−−−−−−→ to CDTs t

µ∗ < ln q + ln 2 → µ <3

2ln(eβ − 1

)+ ln 2

µ∗ <3

2ln(eβ

∗− 1)+ ln 2 → µ <

1

2ln q + ln 2

µ∗ >3

2β∗ + ln q + ln 2 → µ >

3

2ln(q + eβ − 1

)+ ln 2

Table 5.3: Bounds for the critical curve of the q-state Potts model coupled to dual CDTs will

generate bounds for the the critical curve of the q-state Potts model coupled to CDTs.

5.3, parameters (β, µ) and (β∗, µ∗) satisfy the duality relation (5.2).

Tables 5.2 and 5.3 show that to nd bounds for the critical curve for the model on CDTs

provide bounds for the model on dual CDTs, and viceversa. Thus, in the next section we

improves the bounds obtained for the critical curve of the Potts model on CDTs. In aditional,

this approach allows to get a asymptotic behavior of the critical curve for the model on CDTs

and its dual.

58 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.4

5.4 High-T expansion of the Potts model and Proof of

Theorem 5.1.2

Let t be a CDT with periodic boundary condition. The partition function for the Potts

model on t is write in the usual high-T expansion as

ZP (β, q, t) =

(q + h

q

)|E(t)|∑σ

∏〈i,j〉

(1 + fij) (5.35)

where h = eβ − 1 and fij = hq+h

(−1 + qδσi,σj). It can be readily veried that∑

σ fij = 0 for

all i, j ∈ E(t), consequently, all subgraphs with one or more vertices of degree 1 give rise

to zero contributions. Thus, the partition function can be written as follow

ZP (β, q, t) =

(q + h

q

)|E(t)|∑σ

∑A∈G(t)

∏i,j∈A

fij,

where G(t) is the set of families of edges of t without vertices of degree 1. Therefore, we can

rewrite the partition function as

ZP (β, q, t) =

(q + h

q

)|E(t)| ∑A∈G(t)

w(A)

where w(A) =∑σ

∏i,j∈A

fij is a weight factor associated with the subset A. We then pro-

ceeded to determine w(A). An expression of w(A) for general A can be obtained by further

expanding in w(A) the product∑σ

∏i,j∈A

fij. This procedure leads to

w(A) =

(h

q + h

)|A|∑σ

P(A)(σ),

where P(A)(σ) =∏

e∈A(−1 + qδe(σ)), and if e = i, j then δe(σ) = δσi,σj . Expanding

P(A)(σ), we have the following representation

P(A)(σ) = (−1)|A| + (−1)|A|−1q∑e∈A

δe(σ) + (−1)|A|−2q2∑

e1,e2∈A

δe1(σ)δe2(σ)

+ · · ·+ (−1)q|A|−1∑

e1,...,e|A|−1∈A

δe1(σ) . . . δe|A|−1(σ)

+q|A|δe1(σ) . . . δe|A|(σ).

5.4 HIGH-T EXPANSION OF THE POTTS MODEL AND PROOF OF THEOREM 5.1.2 59

Figure 5.4: Examples of three subgraphs of A with 8 edges. It is clear that the term ξ(e1, . . . , e8)depends of the topology of the subgraphs.

We choose k edges e1, . . . , ek of A. These edges form a subgraph of A. Thus, we obtain∑σ

δe1(σ) . . . δek(σ) = q|V (t)|−k+ξ(e1,...,ek)

where ξ(e1, . . . , ek) stands the total numbers of internal faces in each maximal connected

component of e1, . . . , ek (number of independent circuits in e1, . . . , ek). Note that this

terms depends essentially on the topology of e1, . . . , ek (see Figure 5.4). But ξ(e1, . . . , ek) ≤2

3(k + 1) for all k. Thus, we obtain the estimate

∑σ

δe1(σ) . . . δek(σ) ≤ q|V (t)|−k+ 23

(k+1) = q|V (t)|− k3

+ 23

and ∑σ

∑e1,...ek∈A

δe1(σ) . . . δek(σ) ≤(|A|k

)q|V (t)|− k

3+ 2

3 .

60 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.4

Therefore,

∑σ

P(A)(σ) ≤ q|V (t)|+ 23

|A|∑k=0

(|A|k

)(−1)|A|−k( 3

√q2)k = q|V (t)|+ 2

3 ( 3√q2 − 1)|A|,

and

ZP (β, q, t) ≤(q + h

q

)|E(t)|

q|V (t)|+ 23

∑A∈G(t)

(( 3√q2 − 1)

h

q + h

)|A|

≤(q + h

q

)|E(t)|

q|V (t)|+ 23

(1 +

∑k≥1

Ωk(t)uk

),

where Ωk(t) = |A ∈ G(t) : |A| = k| and u = ( 3√q2 − 1)

h

q + h. But Ωk(t) ≤

(|E(t)|k

). Thus,

we get the estimate

ZP (β, q, t) ≤(q + h

q

)|E(t)|

q|V (t)|+ 23 (1 + u)|E(t)|. (5.36)

Proof of Theorem 5.1.2. Using estimate (5.36) and Table 5.1, we write the new bound (5.36)

for the partition function of the Potts model on t in terms of the dynamical variable n(t),

and make similarly computations for the dual case. For that, we consider two cases of interest

separately.

1. The model on CDTs t: In this case, we can to write the new bound for the partition

function in terms of volume n(t) of the triangulation as follow

ZP (β, q, t) ≤(q + h

q

) 32n(t)

q12n(t)+ 2

3 (1 + u)32n(t). (5.37)

Using estimate (5.37), we obtain a new upper bound for the partition function of the q-state

Potts model coupled to CDTs

ΞN(β, µ) ≤ q23

∑t

exp −µn(t) = q23ZN(µ), (5.38)

where µ = µ − 3

2ln

(q + h

q

)− 1

2ln q − 3

2ln(1 + u) and ZN(µ) is the partition function for

pure CDTs (dened in (2.4)) in the cylinder CN with periodical spatial boundary conditions

and for the value of the cosmological constant µ. Hence, the inequality

µ >3

2ln(q + eβ − 1

)+ ln 2− ln q +

3

2ln

(1 + (q2/3 − 1)

eβ − 1

q + eβ − 1

)(5.39)

provides a sucient condition for subcriticality of the q-state Potts model coupled to CDTs

5.4 HIGH-T EXPANSION OF THE POTTS MODEL AND PROOF OF THEOREM 5.1.2 61

(summation is over all Lorentzian triangulation t).

Comparing the new upper bound (5.39) with bounds show in Tables 5.2 and 5.3 for the

model on CDTs, we observe which the condition (5.39) is better than conditions show in

Tables 5.2 and 5.3 for subcriticality behavior of model. Thus, using High-T expansion for

q-state Potts model we get to obtained a better approximation of the critical curve.

Using the duality relation proved in Theorem 5.1.1 and bound (5.39), we obtain a new

condition for subcriticality of the Potts model coupled to dual CDTs

µ∗ >3

2ln(eβ∗ − 1

)+ ln 2 +

3

2ln

(q +

q

eβ∗ − 1

)− 3

2ln q +

3

2ln

(1 +

q2/3 − 1

eβ∗

)

>3

2β∗ + ln 2 +

3

2ln

(1 +

q2/3 − 1

eβ∗

) (5.40)

We will see that this same approach on dual triangulations does not improve the curves

obtained.

2. The model on dual CDTs t∗: Similarly as Eq. (5.37), using Table 5.1 we get the estimate

on a dual triangulation t∗

ZP (β∗, q, t∗) ≤(q + h

q

) 32n(t)

qn(t)+ 23 (1 + u)

32n(t). (5.41)

Using (5.41), we obtain an upper bound for the partition function of the q-state Potts model

coupled to dual CDTs

Ξ∗N(β∗, µ∗) ≤ q23

∑t

exp −µn(t) = q23ZN(µ), (5.42)

where µ = µ∗ − 3

2ln

(q + h

q

)− ln q − 3

2ln(1 + u) and ZN(µ) is the partition function for

pure CDTs in the cylinder CN with periodical spatial boundary conditions and for the value

of the cosmological constant µ. Hence, we obtain the inequality

µ∗ >3

2ln(q + eβ

∗ − 1)

+ ln 2− 1

2ln q +

3

2ln

(1 + (q2/3 − 1)

eβ∗ − 1

q + eβ∗ − 1

), (5.43)

that provide a sucient condition for subcriticality of the Potts model coupled to dual CDTs

(summation is over all dual Lorentzian triangulation t∗)

Finally, using the duality relation proved in Theorem 5.1.1 and bound (5.43), we obtain

62 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.5

a new condition for subcriticality of the Potts model coupled to CDTs

µ >3

2ln(eβ − 1

)+ ln 2 +

3

2ln

(q +

q

eβ − 1

)− 3

2ln q +

3

2ln

(1 +

q2/3 − 1

)

>3

2β + ln 2 +

3

2ln

(1 +

q2/3 − 1

).

(5.44)

It is easy see that (5.43) and (5.44) does not improve the curves obtained in (5.40) and

(5.39), respectively.

Convergence of the Gibbs measure Pβ,µN follows as a corollary. This concludes the proof.

5.5 Connection between transfer matrix and FK repre-

sentation

In this section, we nd a connection between transfer matrix approach and FK repre-

sentation for the Ising model coupled to CDTs, comparing the curve obtained by transfer

matrix approach and the curves obtained by FK representation. In this section we work with

Potts model coupled to DUAL CDTs and will use notations of before chapters.

5.5.1 q = 2 (Ising) systems

The transfer matrix method provides a curve µ∗ = ψ(β∗) (blue line in Figure 5.5), dened

in (4.21), that satisesdψ

dβ∗(0+) = 0.

Therefore, we expect that critical curve satised the same property. Additionally, conditions

tr(KN) < ∞ generate curves µ∗ = γN(β∗) in the quadrant of parameters β∗, µ∗, such that

γN+1 ≤ γN and dγNdβ∗

(0+) = 0 for all N (see Proposition 4.4.3 and 4.4.4 in Chapter 4).

We dene the functions

ϕinf (β∗) = max

2 ln 2,

3

2ln(eβ∗ − 1

)+ ln 2

,

and

ϕsup(β∗) = min

ψ(β∗),

3

2ln(22/3 + eβ

∗ − 1)

+ ln 2

.

Graphs of function ϕinf is a lower barrier for the graph of γN for all N , and the critical

curve of the model. Further, graphs of function ϕsup provides a better upper bound for the

critical curve of Ising model coupled to dual CDTs. Furthermore, in low temperature, the

5.5 CONNECTION BETWEEN TRANSFER MATRIX AND FK REPRESENTATION 63

free energy satisfy

limN→∞

1

Nln ΞN(β∗, µ∗) ≈ ln Λ

(µ∗ − 3

2β∗).

Therefore, maximal eigenvalue Λ of operator K can be approximated by

Λ(β∗, µ∗) ≈ Λ

(µ∗ − 3

2β∗),

where Λ is dened in Chapter 2 in Eq. (2.10).

1 2 3 4

-2

0

2

4

6

8

Figure 5.5: Region where the critical curve of the Ising model coupled to dual CDTs can be located.

5.5.2 q-Potts systems

As in Ising model case, (See Chapter 3), the transfer-matrix formalism suggests rewrite

the partition function as

ΞN(β, µ) = tr KN . (5.45)

where we assume periodic spatial boundary condition and the operator K is dened by

K((t,σ), (t′,σ′)) = 1t∼t′ exp−µ

2(n(t) + n(t′))

(5.46)

× exp−β

2

(h(σ) + h(σ′)

)− βv(σ,σ′)

.

Theorem 3.2.1 and Proposition 3.2.1 given conditions for existence of Gibbs measures

for the model in terms of the trace of operator K.

Estimating tr(KKT): The condition tr(KKT) <∞ guarantees that K and KT are Hilbert-

Schmidt operators. Consequently, the operators K and KT are bounded and K2 and (KT )2

64 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.5

are of trace class. In particular, K2 and (KT )2 belong to space Cp for all p ≥ 2 (see Appendix

A).

By denition the trace (A.2.1), we need to calculate the series

tr(KTK) =∑

(t,σ),(t′,σ′)K2((t,σ), (t′,σ′)). (5.47)

As in Chapter 3, we represent the triangulation t and its supported spin-conguration

σ as

t := (tup, tdo) and σ := (σup,σdo).

Here

tup = (t1up, . . . , tnup), tdo = (t1do, . . . , t

mdo),

and

σup = (σ1up, . . . , σ

nup), σdo = (σ1

do, . . . , σmdo),

assuming that the supporting single-strip triangulation t contains n up-triangles and m

down-triangles. (The actual order of up- and down-triangles and supported spins does not

matter.)

The same can be done for the pair (t′,σ′) (see proof of Lemma 3.2.1). Let recall that the

triangulations t and t′ are consistent (t ∼ t′) i number of the down-triangles in t equals

that of up-triangles in t′.

To calculate the sum (5.47) we divide the summation over (t′,σ′) into a summation over

(t′up,σ′up) and (t′do,σ

′do). Firstly, x a pair (t′up,σ

′up) and make the sum over (t′do,σ

′do). Note

that the term V ((t,σ), (t′,σ′)) depends only on σdo and σ′up. Consequently,

∑t′do,σ′do

K2((t,σ), (t′,σ′)) (5.48)

= e−βH(σ)e−2βV ((t,σ),(t′,σ′))e−µn(t)∑

(t′do,σ′do)

e−βH(σ′)e−µn(t′).

The sum in the right-hand side of (5.48) can be represented in a matrix form. Denote by ek

the unit vectors in Rq: ek = (0, . . . , 1, . . . , 0)T . Next, let us introduce a q× q matrix T where

5.5 CONNECTION BETWEEN TRANSFER MATRIX AND FK REPRESENTATION 65

T = e−µ

eβ 1 · · · 1

1 eβ · · · 1

......

. . ....

1 1 · · · eβ

(5.49)

Denote by n(i), i = 1, . . . , nup(t′) the number of down-triangles in t′ which are between the

ith and (i+ 1)th up-triangles in t′. Let nup(t′) = k then

∑t′do,σ′do

e−βH(σ′)e−µn(t′) =∑

n(i)≥0:∑i n(i)≥1

k∏l=1

(eTσ′lup

T n(l)+1eσ′l+1up

)

=k∏l=1

(eTσ′lup

Meσ′l+1up

)−

k∏l=1

(eTσ′lup

Teσ′l+1up

)(5.50)

where the matrix M is the sum of the geometric progression

M =∞∑n=1

T n (5.51)

Using the same procedure we can obtain the sum over all up-triangles into the triangulation

t. The only dierence is the existence of marked up-triangle in the strip: let as before

nup(t′) = ndo(t) = k then

∑tup,σup

e−βH(σ)e−µn(t) =k−1∏l=1

(eTσlupMeσl+1

up

)(eTσkupM2eσ1

up

)(5.52)

Supposing the existence of the matrixM and using (5.50) and (5.52) we obtain the following:∑tup,σup

∑t′do,σ′do

K2((t,σ), (t′,σ′)) = e−2βV ((tdo,σdo),(t′up,σ′up))

×∑

tup,σup

e−βH(σ)e−µn(t)∑

(t′do,σ′do)

e−βH(σ′)e−µn(t′)

= e−2βV ((tdo,σdo),(t′up,σ′up))

×[ k∏l=1

(eTσ′lup

Meσ′l+1up

) k−1∏l=1

(eTσldoMeσl+1

do

)(eTσkupM2eσ1

up

)−

k∏l=1

(eTσ′lup

Teσ′l+1up

) k−1∏l=1

(eTσldoMeσl+1

do

)(eTσkupM2eσ1

up

)]. (5.53)

66 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.5

Necessary and sucient condition for the convergence of the matrix series for M is that

the maximal eigenvalue of matrix T is less then 1. The eigenvalues of T are

λ1 = e−µ(q + eβ − 1), λ2 = e−µ(eβ − 1), (5.54)

and the above condition means that λ1 < 1 or, equivalently,

µ > ln(q + eβ − 1). (5.55)

Under this condition (5.55), the matrix M is calculated explicitly:

M = g(β, µ) ×

1 + f(β, µ) · · · 1

.... . .

...

1 · · · 1 + f(β, µ)

q×q

, (5.56)

where

g(β, µ) =e−µ

(1− e−µ(eβ − 1))(1− e−µ(q + eβ − 1))

and

f(β, µ) = (eβ − 1)(1− e−µ(q + eβ − 1)).

Now, we express the above sum (5.47) as the partition function of a one-dimensional q-state

Potts model where states are pairs of spins (σldo, σlup) and the interaction is via the matrix

T between the members of the pair and via matrix M between neighboring pairs. More

precisely, introducing the lexicographic order in the set (i, j) : 1 ≤ i, j ≤ q, we dene thefollowing q2 × q2 matrices:

A =

e2β(eTnMel) · (eTmMek) , n = m and l = k

(eTnMel) · (eTmMek) , n 6= m and l 6= k

eβ(eTnMel) · (eTmMek) , either n = m or l = k,

(5.57)

5.5 CONNECTION BETWEEN TRANSFER MATRIX AND FK REPRESENTATION 67

Am =

e2β(eTnMel) · (eTmM2ek) , n = m and l = k

(eTnMel) · (eTmM2ek) , n 6= m and l 6= k

eβ(eTnMel) · (eTmM2ek) , either n = m or l = k,

(5.58)

At =

e2β(eTnTel) · (eTmMek) , n = m and l = k

(eTnTel) · (eTmMek) , n 6= m and l 6= k

eβ(eTnTel) · (eTmMek) , either n = m or l = k,

(5.59)

Atm =

e2β(eTnTel) · (eTmM2ek) , n = m and l = k

(eTnTel) · (eTmM2ek) , n 6= m and l 6= k

eβ(eTnTel) · (eTmM2ek) , either n = m or l = k.

(5.60)

Now for the sum under consideration (5.47) we obtain using representation (5.53)

tr(KTK) =∑

(t,σ),(t′,σ′)K2((t,σ), (t′,σ′)) = tr

(( ∞∑k=0

Ak)Am)− tr

(( ∞∑k=1

Akt)Atm

).

By the construction the matrix A is greater then At elementwise. Thus the eigenvalue of

matrixA is greater than the eigenvalue of the matrixQt (it follows from the Perron-Frobenius

theorem). Therefore the necessary and sucient condition for the convergence in (5.47) is

that the largest eigenvalue of A is less than 1. In general, it is impossible to calculate its

eigenvalue analytically. For case q = 2 (Ising model) was possible calculated its eigenvalue

(see Chapter 3 for review), but q > 2 it very dicult. In the case of q = 4, we make a

comparison between curves obtained by duality relation and high-T expansion in Section

5.4, and curve obtained using numerical simulation. See below graph.

Finally as byproduct of Theorem 5.1.2, we have the following assertions for the free

energy for q-state Potts model.

Corollary 5.5.1. The free energy for q-state Potts model coupled to dual CDTs satisfy

limN→∞

1

Nln ΞN(β∗, µ∗) ≤ ln Λ

(µ∗ − 3

2β∗ − r(β∗)

),

68 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION 5.6

Figure 5.6: The blue line is the simulation of ||A||2 = 1 for q = 4. Black line: µ∗ = 3 ln 2. Green

line: µ∗ = 32 ln

(eβ∗ − 1

)+ ln 2. Red line: µ∗ = 3

2 ln(42/3 + eβ

∗ − 1)+ ln 2.

where r(β∗) =3

2ln

(1 +

q2/3 − 1

eβ∗

). Moreover, in low temperature, we have that

limN→∞

1

Nln ΞN(β∗, µ∗) ≈ ln Λ

(µ∗ − 3

2β∗).

Therefore, maximal eigenvalue Λ of operator K can be approximated by

Λ(β∗, µ∗) ≈ Λ

(µ∗ − 3

2β∗),

where Λ is dened in Chapter 2 in Eq. (2.10).

5.6 Discussion and outlook

This chapter we dene and study a Potts model coupled to CDTs and dual CDTs employ-

ing FK representation and duality on graphs. The results obtained serve for the Ising model

(see Section 5.5.1). In particular, we nd a better region where the free energy there exists

and can be extended analytically (line I and I ′ in Figure 5.1), and this region depend on

analyticity of maximal eigenvalue Λ of operator K and eigenvalue Λ, dened in Eqn (2.10).

This remark permite us to give the conjecture that the boundary of the critical domain

coincides with the locus of points (β, µ) where Λ loses either the property of positivity or

the property of being a simple eigenvalue.

Notice that, if (β, µ) satisfy hypothesis of Theorem 5.1.2, we don't have information on

either K belong to Cp or not, for some p > 2. This issue needs a further study.

We can give tha follos assertion: If (β, µ) satisfy the subcritical behavior of Theorem 5.1.2,

the limiting probability distribution Pβ,µ is represented by a positive recurrent Markov chain

DISCUSSION AND OUTLOOK 69

with states (t,σ) as Theorem 3.2.2. In the subcritical region of Theorem 5.1.2, the typical

triangulation for annealed Potts model coupled to CDTs is the same as subcritical case in

pure CDT (see Theorem 2.2.2).

The new bounds for the critival curve for arbitrary q suggests that the Potts model

coupled to CDTs exhibit a phase transition only on the critical curve and a rst-order

singularity at a unique point βcr ∈ (0,∞). Additionally, the triangulations in the annealed

model exhibit critical behavior as critical case in pure CDT (see Theorem 2.2.2) only on the

critical curve. This direction also requires further research.

70 POTTS MODEL COUPLED TO CDTS AND FK REPRESENTATION

Appendix A

The von Neumann-Schatten Classes of

Operators

This appendix in concerned with certain classes Cp (1 ≤ p < ∞) of linear operators.

on a Hilbert space H. These classes are important for to study the transfer operator in

statistical mechanics because that operator encodes information and to study the behavior

of the statistical mechanics system.

A.1 The space Cp and rst properties

In this section we dene the space Cp and given some properties.

Denition A.1.1. When 1 ≤ p <∞, Cp is the set of all operators T in B(H) which satisfy

the following condition: for each orthonormal system φk : k ∈ K in H,∑k∈K

|〈Tφk, φk〉|p <∞.

We shall adopt the convention that C∞ is B(H). Each Cp is a linear subspace of B(H)

and Cp ⊆ Cq if 1 ≤ p ≤ q ≤ ∞. We can to see that T ∗ ∈ Cp (adjoint operator) whenever

T ∈ Cp, and that, if 1 ≤ p <∞, then each element of Cp is a compact operator.

Lemma A.1.1. Suppose that 1 ≤ p < ∞, T is a compact self-adjoint operator on H, andλn is the sequence of non-zero eigenvalues of T , counted according to their multiplicities.

1. If T ∈ Cp, then∑

n |λn|p <∞.

2. If∑

n |λn|p <∞, then T ∈ Cp and, for each orthonormal system φk : k ∈ K in H,∑k∈K

|〈Tφk, φk〉|p <∑n

|λn|p.

71

72 APPENDIX A

A.2 The trace class C1

In this section we will dene the trace of a operator on the class C1.

Lemma A.2.1. Let T ∈ C1 and suppose that φk : k ∈ K is an orthonormal basis in H.

1. The sum∑

k〈Tφk, φk〉 exist, and does not depend on the particular choise of the or-

thonormal basis φk : k ∈ K.

2. If T = T ∗, then∑

k〈Tφk, φk〉 =∑

k λk, where λk is the sequence of non-zero eigen-

values of T , counted according to their multiplicities.

Denition A.2.1. The ideal C1 in B(H) is called trace class of operators on H. If T ∈ C1

and φk : k ∈ K is an orthonormal basis in H, then the trace of T , denoted by tr(T ), is

dened by the equation

tr(T ) =∑k

〈Tφk, φk〉.

Lemma A.2.1 shows that tr(T ) depends only on T (not on the choice of the orthonormal

basis), and that tr(T ) is the sum of the eigenvalues of T when T = T ∗.

The main algebraic properties of tr are the following.

Theorem A.2.1. Suppose that S, T ∈ C1, A ∈ B(H) and α, β are scalars.

1. tr(αS + βT ) = αtr(S) + βtr(T ).

2. tr(S∗) = tr(S).

3. tr(S) > 0 if S > 0.

4. tr(AS) = tr(SA).

The main result of this section, used in Chapter 3, is the following.

Theorem A.2.2. Suppose that T is a trace class operator acting on a Hilbert space H,and λk is the sequence of non-zero eigenvalues of T , counted according to their algebraic

multiplicities. Then

tr(T ) =∑k

λk. (A.1)

A.3 The Banach space CpSuppose that T is a compact operator acting onH, and denote by VTHT the polar decom-

position of T . Then T = VTHT and HT = (T ∗T )1/2. Remember that VT is a partial isometry

on the closed range RH of H. It is well know that there exist a decreasing sequence µn of

THE BANACH SPACE CP 73

positive real numbers (the eigenvalues of HT , counted according to their multiplicities), and

orthonormal sequence φn, ψn, such that

HT (x) =∑n

µn〈x, φn〉φn,

T (x) =∑n

µn〈x, φn〉ψn.

Given p ≤ 1 the function fp(t) = tp is continuous on the non-negative real axis, and hence

also on the spectrum of the positive operator HT . The operator fp(HT ) will be denoted by

HpT . The operator H

pT is compact and can be represented by

HpT (x) =

∑n

µpn〈x, φn〉φn.

Lemma A.3.1. Suppose 1 ≤ q ≤ p <∞ and T ∈ B(H). Then the following three conditions

are equivalent

1. T ∈ Cp,

2. HT ∈ Cp,

3. Hp/qT ∈ Cq.

From Lemma A.3.1 and the equivalence of conditions (i) and (ii) in the before lemma,

it follows that a compact operator T on H lies Cp if only if the sequence µn of non-zeroeigenvalues of HT = (T ∗T )1/2 satises

∑n µ

pn <∞.

Denition A.3.1. Suppose 1 ≤ p <∞ and T ∈ Cp. Then, we dene

||T ||p = [tr(HpT )]1/p =

(∑n

µpn

)1/p

.

It is not immediately obvious that || · ||p is a norm on Cp. Since ||T || = ||HT || = µ1

(maximal eigenvalue), we have

||T || ≤ ||T ||p, for T ∈ Cp.

Lemma A.3.2. For each T ∈ C1, |tr(T )| ≤ ||T ||1.

Lemma A.3.3. For each T ∈ Cp, ||T ∗||p = ||T ||.

Lemma A.3.4. Suppose 1 ≤ p < ∞, T ∈ Cp, and λn is the sequence of non-zero eigen-

values of T , counted according to their algebraic multiplicities. Then(∑n

|λn|p)1/p

≤ ||T ||p.

74 APPENDIX A

A.4 The Hilbert-Schmidt class

Denition A.4.1. The ideal C2 in B(H) is called the Hilbert-Schmidt class of operators on

H.

If VTHT is the polar decomposition of an element T of C2, then

(tr(T ∗T ))1/2 = (tr(H2T ))1/2 = ||T ||2.

Theorem A.4.1. Suppose that T ∈ B(H), φk and ψk are orthonormal bases in H. Thethe following three conditions are equivalent.

1.∑

k ||Tφk||2 <∞,

2.∑

j,k |〈Tφj, ψk〉|2 <∞,

3. T ∈ C2.

When these conditions are satised, the sums occurring in (i) and (ii) are both equal to

||T ||22.

Appendix B

Krein-Rutman theorem

The Krein-Rutman theorem plays a very important role in linear fuctional analysis, as

it provides the abstract basis for the proof of the existence of various principal eigenvalues,

which in turn are crucial in transfer matrix formalism of statistical mechanics system (and

another areas as nonlinear partial dierential equations, bifurcation theory, etc). In this

appendix, we will give the well-known Krein-Rutman theorem.

B.1 Krein-Rutman Theorem and the Principal Eigen-

value

Let X a Banach space. By cone K ⊂ X we mean a closed convex set such that λK ⊂ K

for all λ ≥ 0 and K ∩ (−K) = 0. A cone K in X induce a partial ordering ≤ by the rule:

u ≤ v if and only if v − u ∈ K. A Banach space with such an ordering is usually called

a partially ordered Banach space and the cone generating the partial ordering is called the

positive cone of the space. If K −K = X, i.e., the set u − v : u, v ∈ K is dense in X,

then K is called a total cone. If K −K = X, K is called a reproducing cone. If a cone has

nonempty interior Ko, then it is called a solid cone. Any solid cone has the property that

K − K = X, in particular, it is total. We write u > v if u − v ∈ K \ 0, and u v if

u− v ∈ Ko.

The main results of this appendix, used in Chapter 3, are the following.

Theorem B.1.1 (The Krein-Rutman Theorem [KR48]). Let X a Banach space, K ⊂ X a

total cone and T : X → X a compact linear operator that is positive, i.e., T (K) ⊂ K, with

positive spectral radius r(T ). Then r(T ) is an eigenvalue with an eigenvector u ∈ K \ 0:Tu = r(T )u. Moreover, r(T ∗) = r(T ) is an eigenvalue of T ∗.

Let us now use Theorem B.1.1 to derive the following useful result.

Theorem B.1.2. Let X a Banach space, K ⊂ X a solid cone, T : X → X a compact linear

operator which is strongly positive, i.e., Tu 0 if u > 0. Then

75

76 APPENDIX B

1. r(T ) > 0, and r(T ) is a simple eigenvalue with an eigenvector v ∈ Ko; ; there is no

other eigenvalue with a positive eigenvector.

2. |λ| < r(T ) for all eigenvalues λ 6= r(T ).

Let us recall that λ is a simple eigenvalue of T if there exists v 6= 0 such that Tv = λv

and (λI − T )nw = 0 for some n ≥ 1 implies w ∈ spanv.

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