Properties of self-avoiding walks and a stress-energy tensor in the O(n) model

Thesis

Reference

Properties of self-avoiding walks and a stress-energy tensor in the O(n) model

GLAZMAN, Alexander

Abstract

This thesis is devoted to the study of 2-dimensional models of statistical mechanics. More precisely, we focus on the loop O(n) model and two classical models which can be realized as its particular cases: the Lenz-Ising model (n = 1) and the self-avoiding walk (n = 0). The main goals are to extend our knowledge about these models and to better understand their connection with the Conformal Field Theory which is conjectured to describe the scaling limits.

The results known earlier for the self-avoiding walk on the hexagonal lattice are extended to the self-avoiding walk with integrable weights. A discrete stress-energy tensor in the loop O(n) model is constructed and shown to converge to its continuous counterpart for the Ising model.

The endpoint of the self-avoiding walk is shown to be delocalized. The main tools used in the thesis are (para)fermionic observable, Yang-Baxter equation and Kesten's pattern lemma.

GLAZMAN, Alexander. Properties of self-avoiding walks and a stress-energy tensor in the O(n) model. Thèse de doctorat : Univ. Genève, 2016, no. Sc. 4932

URN : urn:nbn:ch:unige-877293

DOI : 10.13097/archive-ouverte/unige:87729

Available at:

http://archive-ouverte.unige.ch/unige:87729

Disclaimer: layout of this document may differ from the published version.

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Universit´e de Gen`eve Facult´e des Sciences

Section de Math´ematiques Professeur Stanislav Smirnov

Properties of self-avoiding walks and a stress-energy tensor in the O(n) model.

Th`ese

pr´esent´ee `a la Facult´e des Sciences de l’Universit´e de Gen`eve pour l’obtention du grade de Docteur `es sciences, mention math´ematiques

par

Alexander Glazman

de

Saint-P´etersbourg (Russie)

Th`ese N^◦ 4932

Gen`eve Atelier d’impression ReproMail de l’Universit´e de Gen`eve 2016

Abstract

This thesis is devoted to the study of 2-dimensional models of statistical mechanics. More precisely, we focus on the loop O(n) model and two classical models which can be realized as its particular cases: the Lenz-Ising model (n = 1) and the self-avoiding walk (n = 0).

The main goals are to extend our knowledge about these models and to better understand their connection with the Conformal Field Theory which is conjectured to describe the scaling limits. We start by introducing the loop O(n) model on the hexagonal lattice, giving a brief account on the recent progress in the rigourous understanding of its properties. We then state our main results which are proven in the rest of the thesis. The tool which allowed several breakthroughs in the last decade is the (para)fermionic observable introduced by Smirnov.

This observable stands at the core of the argument in Chapters 2 and 3.

In Chapter 2, we consider the self-avoiding walk on the dual Z² lattice with integrable weights. This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. Nien- huis found a family of integrable weights parametrized by angle θ∈[^π₃,^2π₃ ], which satisfy the Yang-Baxter equation. We discuss the asymptotic of the partition function of these walks and, by means of the Yang-Baxter transformation, show that the 2-point function in the half-plane does not depend on the rhombic tiling. In fact, this statistic coincides with the one of the self-avoiding walk on the hexagonal lattice. From this we derive that, for any rhombic tiling, the partition function of self-avoiding bridges of a fixed height vanishes as the height tends to infinity, generalizing a statement that was known previously only for the hexagonal lattice. In the latter case we give a short proof using the parafermionic observable.

In Chapter 3, we study the loop O(n) model on the honeycomb lattice. The integrable weights discovered by Nienhuis allow to local infinitesimal deformations of the lattice. Tak- ing a derivative of the partition function under these deformations, we construct a discrete stress-energy tensor. For n ∈ [−2,2], it gives a new observable satisfying a part of the Cauchy-Riemann equations. We conjecture that it is approximately discrete-holomorphic and converges to the stress-energy tensor in the continuum, which is known to be a holo- morphic function with the Schwarzian conformal covariance. In support of this conjecture, we prove it for the case of n = 1 which corresponds to the Ising model. Moreover, in this case, we show that the correlations of the discrete stress-energy tensor with primary fields converge to their continuous counterparts, which satisfy the OPEs given by the CFT with central charge c = 1/2. Proving the conjecture for other values of n remains a challenge.

In particular, this would open a road to establishing the convergence of the interface to the corresponding SLEκ in the scaling limit.

In Chapter 4, we prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d ≥ 2. We show that the probability that a walk of length n ends at a pointx tends to 0 asntends to infinity, uniformly inx. Also, whenxis fixed, with

||x||= 1, this probability decreases faster thann^−1/4+ε for any ε >0. This provides a bound on the probability that a self-avoiding walk is a polygon.

Cette th`ese est consacr´ee `a l’´etude des mod`eles 2-dimensionnels de m´ecaniques statistiques.

Plus pr´ecis´ement, nous nous concentrons sur le mod`ele de boucleO(n) et sur deux mod`eles classiques qui peuvent ˆetre r´ealis´es comme cas particuliers de ce dernier : le mod`ele de Lenz- Ising et la marche auto-´evitante. Les buts principaux sont d’´etendre nos connaissances `a propos de ces mod`eles et de mieux comprendre leur relation avec la th´eorie des champs conformes, qui d´ecrit conjecturalement limite d’´echelle. Nous commen¸cons par introduire le mod`ele de boucleO(n) sur le r´eseau hexagonal, en donnant un bref compte-rendu des r´ecents progr`es dans la compr´ehension de ses propri´et´es. Nous ´enon¸cons ensuite les r´esultats princi- paux, qui seront prouv´es dans le reste de la th`ese. L’outil ayant permis de nombreuses perc´ees pendant cette derni`ere d´ecennie est l’observable (para)fermionique , introduit par Smirnov.

Cette observable est au coeur des d´emonstrations dans les chapitres 2 et 3.

Dans le chapitre 2, nous consid´erons la marche auto-´evitante sur le r´eseau dual Z² avec poids int´egrables. Cette marche peut traverser le mˆeme carr´e deux fois mais ne peut pas traverser la mˆeme arˆete plus d’une fois. Le poids de chaque carr´e d´epend de la mani`ere dont la marche le traverse et le poids total de la marche est calcul´e comme le produit de ces poids. Nienhuis a d´ecouvert une famille de poids int´egrables, param´etris´ee par un angleθ∈[π/3,2π/3], qui satisfont l’´equation de Yang-Baxter. Nous discutons l’asymptotique de la fonction de partition de ces marches et, en utilisant la transformation de Yang-Baxter, montrons que la fonction `a 2 points dans le demi-plan ne d´epend pas du pavage rhombique.

En fait, cette statistique co¨ıncide avec celle de la marche auto-´evitante sur le r´eseau hexa- gonal. De cela nous concluons que pour tout pavage rhombique, la fonction de partition des ponts auto-´evitants d’une hauteur fix´ee tend vers z´ero quand la hauteur tend vers l’infini, g´en´eralisant ainsi un r´esultat pr´ec´edemment connu seulement pour le r´eseau hexagonal. Dans ce dernier cas, nous donnons une courte preuve utilisant l’observable parafermionique.

Dans le chapitre 3, nous ´etudions le mod`ele de boucleO(n) sur le r´eseau hexagonal. Les poids int´egrables d´ecouverts par Nienhuis permettent des d´eformations infinit´esimales locales du r´eseau. En prenant la d´eriv´ee de la fonction de partition d´eform´ee de cette fa¸con, nous construisons un tenseur ´energie-impulsion discret. Pourn∈[0,2] cela donne une nouvelle ob- servable qui satisfait partiellement les ´equations de Cauchy-Riemann equations. Nous conjec- turons que cette observable est approximativement discrete-holomorphe et converge vers le tenseur ´energie-impulsion, dont on sait qu’il est holomorphe avec la covariance Schwarzienne conforme. Pour appuyer cette conjecture, nous la d´emontrons dans le cas n=1 qui correspond au mod`ele d’Ising. De plus, dans ce cas, nous montrons que les corr´elations de ce tenseur impulsion-´energie discret avec des champs primaires convergent vers leurs versions continues, qui satisfont les OPEs donn´ees par le CFT de charge centrale c=1/2. D´emontrer la conjecture pour d’autres valeurs denreste un probl`eme ouvert. Cela donnerait une possibilit´e d’´etablir la convergence de l’interface vers la SLE_κ correspondante dans la limite d’´echelle.

Dans le chapitre 4, nous d´emontrons deux r´esultats sur la d´elocalisation de l’extr´emit´e d’une marche auto-´evitante uniforme surZ^d pour d ≥ 2. Nous montrons que la probabilit´e qu’une marche de longueurnse termine `a un pointx tend vers 0 lorsquentend vers l’infini, uniform´ement en x. De plus, lorsque x est fix´e avec ||x||= 1, cette probabilit´e diminue plus rapidement quen^−1/4+ε. Cela donne une borne sur la probabilit´e qu’une marche auto-´evitante soit un polygone.

Acknowledgments.

First of all I want to thank my advisor Stanislav Smirnov, who introduced me to this very active and fantastic field of mathematics and who was guiding me through it for five years. I will be forever grateful for the amount of time he spent with me, especially in the beginning when it was much needed. His enthusiasm in challenging hard problems is very inspiring and extends beyond mathematics. I was benefitting greatly from having him not only as a PhD advisor but also as a mentor, teaching me many useful things besides mathematics as well.

I feel very lucky to have collaborated and frequently discussed with Dmitry Chelkak, who I consider a very strong mathematician, a great person and a saviour throughout various difficult situations along the way. He as well became a mentor to me, providing me with lots of however small or big always very exact, advice, which I am trying to follow as much as I can. Thank you Dmitry for all your guidance, help and support.

I would like to thank Hugo Duminil-Copin, Cl´ement Hongler, Kostya Izyurov and Antti Kemppainen for being such supportive elder academic brothers. I was fortunate to work with Hugo and Cl´ement where the former taught me a lot about the self-avoiding walk and the latter about the conformal field theory. Special thanks to Hugo for agreeing to be in the jury.

Kostya was one of the first people from the field I met during my studies and he was very kind to answer all the questions I asked him. While riding the magnificent ski lifts of Switzerland, I had several very dear to me discussion with Antti.

I am thankful to Ioan Manolescu and Alan Hammond for an interesting collaboration.

Special thanks to Ioan for sharing his expertise in numerous aspects of life.

I would also like to thank Itai Benjamini and Yvan Velenik for several inspiring discussions and for agreeing to be in the jury.

Thanks to everybody who came to share their thoughts with me after my talks, thus helping me to improve them further. In particular, I would like to thank David Cimasoni who once gave me a long list of very precious tips, which I now can only hope he does not remember anymore.

To work in a such a big and active body as the Geneva probability group was a great experience and I would like to thank everyone who was part of it in the last 5 years for creating such an unique and inspiring ambiance.

Before they feel left out and get cranky, I would like to thank all my friends in Geneva, St Petersburg and the rest of this world, for their constant support, love and laughter.

Last but not least, I would like to distinctly thank my dear family, especially my parents Liudmila and Lev, for everything they did for and will always mean to me.

Contents

Contents 7

1 Introduction 9

1.1 LoopO(n) model. Definitions, conjectures and known results . . . 12

1.2 Self-avoiding walk . . . 14

1.3 Main results. . . 15

1.3.1 Self-avoiding walk onZ² with integrable weights . . . 15

1.3.2 Infinitesimal deformations of the discrete complex structure and the discrete stress-energy tensor . . . 19

1.3.3 Convergence results for the Ising model . . . 21

1.3.4 Delocalisation of the endpoint of the self-avoiding walk. . . 24

1.4 Organisation of the proofs . . . 26

2 Weighted self-avoiding walk onZ² 29 2.1 Caseθ= ^π₃ and the sketch of the proof. . . 29

2.2 Parafermionic observable and integrable weights . . . 31

2.3 Proofs of Theorems 1-2 . . . 33

2.4 Self-avoiding bridges and the 2-point function. . . 37

2.5 Critical weights for the loopO(n) model. . . 42

3 Discrete stress-energy tensor in the loop O(n) model 45 3.1 Definition of the discrete stress-energy tensor in the loopO(n) model . . . 45

3.1.1 The loop O(n) model on discrete Riemann surfaces. . . 45

3.1.2 Infinitesimal deformations and definition ofTedge and Tmid. . . 48

3.1.3 Alternative definition ofTedge and Tmid. . . 49

3.1.4 Local relations on Tedgeand Tmid.. . . 52

3.1.5 Complex-valued observable T . . . 54

3.1.6 Infinitesimal deformations in terms of conical singularities . . . 55

3.2 The casen= 1: Ising model on faces of the honeycomb lattice . . . 56

3.2.1 Stress-energy tensor as a local field . . . 56

3.2.2 Stress-energy tensor expectations via fermionic observables . . . 60 7

3.2.3 Dobrushin boundary conditions and two-point expectations via four-

point fermionic observables . . . 62

3.2.4 Spinor observables and correlations with the spin field . . . 65

3.3 Ising correlation functions in continuum . . . 67

3.3.1 Fermionic correlators as solutions to boundary value problems for holo- morphic functions . . . 67

3.3.2 Stress-energy tensor and energy density correlations . . . 69

3.3.3 Correlations with the spin field . . . 71

3.3.4 Schwarzian conformal covariance and singularities of correlation functions 73 3.4 Convergence results for the Ising model . . . 75

3.4.1 S-holomorphicity and convergence of discrete fermionic observables . . 76

3.4.2 Proofs of Theorems 5–7 . . . 78

3.4.3 Convergence of the spinor observables . . . 84

A Appendix . . . 88

A.1 Coefficients in the definition ofTedge and Tmid . . . 88

A.2 S-holomorphic functions on the honeycomb grid and the construction of the full-plane observable . . . 91

4 On the probability that self-avoiding walk ends at a given point 97 4.1 Preliminaries . . . 97

4.1.1 General notation . . . 97

4.1.2 The multi-valued map principle . . . 97

4.1.3 Unfolding self-avoiding walks . . . 98

4.1.4 The Hammersley-Welsh bound . . . 99

4.2 The shell of a walk: definition and applications . . . 99

4.2.1 Shell probabilities are stable under perturbation of walk length . . . . 100

4.2.2 Redistribution of patterns . . . 102

4.3 Delocalization of the endpoint . . . 105

4.3.1 The case of bridges . . . 106

4.3.2 The case of half-space walks . . . 107

4.3.3 The case of walks (proof of Theorem 9) . . . 108

4.4 Quantitative decay for the probability of ending atx . . . 111

4.4.1 An overview of the proof of Theorem 8. . . 112

4.4.2 Proof of Theorem 8 . . . 113

Bibliography 121

Chapter 1

Introduction

A general idea in the field of lattice models is to describe a certain macroscopic phe- nomenon from physics or chemistry using its discrete approximation at the microscopiclevel.

An example coming from physics is an effect of ferromagnetism. It is well-known that each ferromagnetic metal has its own Curie temperature above which it stops being ferromagnetic.

The Lenz-Ising model describes this phenomenon of a phase transition and is arguably the most studied model in mathematical physics. Each configuration is a particular way to as- sign +1 and −1 to the sites of a fixed finite part of a lattice; the Hamiltonian is given by the number of adjacent sites with different signs. The original motivation was that spins correspond to magnetic moments of individual atoms, simplified to take any two possible orientations. It was shown by Peierls [Pei36] that in the dimension 2 and bigger, in low tem- perature spins are highly influenced by the exterior magnetic field and they get co-orientated, whilst in high temperature orientation of the spins in the bulk is chaotic. This explains why the magnetic properties of a metal depend on the temperature. Since the work of Peierls, there was a lot of progress in the investigation the the Ising model at the critical temperature, especially in two dimensions. Below we give more details about this.

Another famous model is the self-avoiding walk, the model of a polymer proposed by Flory and Orr [Flo53,Orr47]. In this case each configuration is a non self-intersecting path in a graph starting from a fixed vertex. These walks turned out to be a very interesting object leading to rich mathematical theories, see [MS93,BDCGS11], and raising important challenges (the self-avoiding walk is difficult to understand because of its non-markovity). There are not so many rigorous statements in the dimension 2, and Flory’s conjectures about the number of the self-avoiding walks of a fixed length and about the mean-square displacement are still far out of reach.

On the hexagonal lattice both Ising model and a version of the self-avoiding walk can be described by the loop O(n) model. In this model the configurations are even subgraphs of the hexagonal lattice. Each connected component of such a subgraph is a loop and the weight is proportional to x^#edgesn^#loops, where x is an edge-weight and n is a loop-weight.

The value n = 1 corresponds to the Ising model as each even subgraph defines two spin±1 configurations on the faces of the hexagonal lattice such that edges of the subgraph separate adjacent spins of a different sign. To get the self-avoiding walk one needs to take n= 0 and make two chosen points on the boundary to have degree 1; this implies absence of loops and existence of a simple walk connecting these two points.

9

One can consider a sequence of refining lattice approximations of a fixed finite domain on the plane. For the loop O(n) models, it is conjectured that at the critical point there exists a limit of this sequence of probability measures as the mesh size of the lattice tends to 0.

Moreover, it is expected that the limiting probability measure does not depend on a choice of approximations and is conformally invariant; a measure PΩ is calledconformally invariant if it is invariant under conformal mappings, i.e. PΩ⁰ =PΩ◦ϕ for any conformal ϕ: Ω⁰ → Ω (a mapping is conformal if it preserves angles).

The conformal invariance turns out to be an extremely strong property. Let us restrict our attention to random curves, i.e. measures supported on non self-intersecting curves starting at a fixed point on the boundary of a simply connected domain. We say that a random curve satisfies domain Markov property if it cannot distinguish its prior trajectory from the boundary, i.e. given the curve up to time tthe law of the rest of it is the same as the law of the curve starting at γ(t) in Ω\γ[0, t].

A big breakthrough occurred in 2000when Schramm [Sch00] proved that all conformally invariant random curves satisfying domain Markov property can be described by a one param- eter family of measures. This family of measures is now called SLE(κ) (Schramm-Loewner evolution) and has a direct construction in terms of Loewner chains with√

κB_tas the driving function, whereB_t is a Brownian motion.

Since then, one of the biggest conjectures in two-dimensional lattice models is that, at criticality, interfaces converges to SLEs in the scaling limit. The exact value ofκcan be derived from physics predictions and SLE computations. For the loop O(n) model, an interface is conjectured to converge to SLE(κ) forκ= 4π/(2π−arccos(−n/2)). Convergence to SLE(κ) was proven for several models: site percolation on the triangular lattice (κ = 6) by Smirnov in 2001 [Smi01], uniform spanning tree (κ = 8) and loop-erased random walk (κ = 2) by Lawler, Schramm and Werner in 2003 [LSW04b], harmonic explorer (κ = 4) by Schramm and Sheffield, Ising model (κ = 3) and FK-Ising model (κ = 16/3) by Smirnov et al in 2007-2013 [Smi10a,CS12,KS12,CDCH13,CDCH⁺14].

In all these seminal papers a particulardiscrete martingale observable stands at the core of the proof. This observable is a function on edges/sites/etc of the lattice that satisfies par- ticular local relations (discrete harmonicity/holomorphicity) and is a martingale with respect to the growing interface. In some sense, this allows to encode the interface by this observ- able and any information derived about the latter casts a light upon the former. Moreover, even a martingale observable satisfying only a part of the relations can be an important tool.

Probably, the best and the most relevant example is the recent computation of the connective constant for the self-avoiding walk on the hexagonal lattice (the rate of the growth of the num- ber of simple paths of a fixed length), due to Duminil-Copin and Smirnov in 2010 [DCS12b].

The main tool in their proof is the so-called parafermionic observables that satisfies a part of discrete Cauchy-Riemann equations. It allows to get a relation on the partition function of self-avoiding walks. Slightly abusing the notion of integrability we will call such walks integrable.

Below is the list of main questions which we are addressing in the current thesis, together with short answers and references to the places in the text, where these questions are discussed in more details

Question 1. Are there other integrable models of self-avoiding walks and can one study there properties using the parafermionic observable, introduced in [Smi06]?

11 The answer is positiv — one can consider self-avoiding walks on the square lattice, where a walk is allowed to touch itself but gets penalised for this. A one parameter family of integrable weights satisfying the Yang-Baxter relation was discovered by Nienhuis [Nie90]. Using the same technique as in [DCS12b], we compute the analogue of the connective constant in this case (Theorems 1 and 2). Similar ideas and the Yang-Baxter relation imply that the 2- point function in the half-plane in this case is the same as for the self-avoiding walk on the hexagonal lattice (Theorem 3) and that the partition function of bridges of a fixed width tends to 0 as the width tends to infinity (Theorem4). The latter was known for the hexagonal lattice [BBMdG⁺14]. In this case, we provide a new short proof which might be of independent interest (Proposition 2.4.2).

The preliminaries and main results are given in Section1.3.1and the proofs can be found in Chapter 2. Our exposition follows [Gla15] and [GM].

Question 2. Can one consider other discrete martingale observable which would give us some new information about other integrable models?

By means of local deformation of the lattice, we construct a new discrete martingale observable for the loop O(n) model, which satisfies a half of the discrete Cauchy-Riemann relations (section 3.1). Even in the case of the Ising model (n = 1), the other half of the relations is missing, and unfortunately, we do not know how to use this observable to derive any information about the loop O(n) model or other models. In Section 1.1 we formulate a conjecture about the convergence of this observable to a holomorphic with a Schwarzian conformal covariance. Proving this conjecture would result in establishing the convergence of the interface in the loop O(n) model to the corresponding SLE(κ).

Question 3. Is it possible to prove the convergence of this observable at least in some cases?

Do these observables have a meaning in terms of the conformal field theory?

We prove our conjecture for the Ising model (n= 1), and identify the limit of correlations of our observable and other fields (spin, energy density). Moreover, we show that these limits have the same conformal covariance and satisfy the same OPEs (operator product expansion) as the corresponding quantities for the stress tensor from the conformal field theory. Thus, we call this observable a discrete stress-energy tensor for the loop O(n) model.

The list of results is given in the Section1.3.3, the proofs are provided in Chapter3. Our exposition follows [CGS16].

Question 4. What can one prove for the self-avoiding walk onZ², where there is no observ- ables/integrability?

Itai Benjamini asked if one can prove that the probability that the self-avoiding walk of a fixed length ends right next to the origin vanishes as the length tends to infinity. Denoting byn the length of the walk, we prove that for anyε >0 for n large enough, this probability is bounded above byn^−1/4+ε (Theorem 8). We also show that the probability that the self- avoiding walk of a fixed length ends at given point vanishes as the length tends to infinity (Theorem9).

The list of results is given in Section1.3.4, the proofs are can be found in Chapter4. The material provided in this chapter closely follows [DCGHM16].

1.1 Loop O(n) model. Definitions, conjectures and known re- sults

The loop O(n) model is one of natural generalisations of the Ising model introduced by Lenz in [Len20]. His student Ising showed in his thesis [Isi] that in one dimension the Lenz-Ising model exhibits no phase transition, assuming this to be the case in higher di- mensions as well. Though phase transition in dimension d ≥ 2 was eventually established by Peierls [Pei36], the misunderstanding led to the introduction of other models, including the Heisenberg model [HK34], with spins taking values on the unit circle. The latter was generalized [Sta68] to the spinO(n) model where the spins take values on the unit sphere in the n-dimensional space, n ∈ Z+. Besides the Heisenberg model for n = 2, it contains the Ising model for n= 1 as well. The spinO(n) model turned out to be very difficult to analyze, the main conjecture being the absence of the phase transition for n≥3, see [Kup80] for the list of known results. On the hexagonal lattice the spinO(n) model was conjectured to be in the same universality class as the loopO(n) model introduced in [DMNS81], which is defined for all n ∈ R+, with some quantities making sense even for negative values of n. For more details on the relation between the two models and historical remarks see [DCPSS14], where the case of a largenis considered. We are in position to define the loop O(n) model on the hexagonal lattice Hex_δ, where δ is the edge length.

Given a simply-connected domain Ω, we will use the following notation, see fig.1.1:

— Fδ(Ω) denotes the set of faces of Hex_δ entirely contained in Ω;

— Eδ(Ω) denotes the set of edges contained in boundaries of faces inFδ(Ω);

— Vδ(Ω) denotes the set of vertices of Hex_δ incident to edges inEδ(Ω);

— ∂Vδ(Ω) denotes the set of vertices ofVδ(Ω) incident to exactly 2 edges inEδ(Ω);

— Ω_δ denotes the subgraph of Hex_δ defined by Vδ(Ω) and Eδ(Ω) plus the half-edges incident to ∂Vδ(Ω).

We omit indices δ everywhere before we start speaking about the convergence as δ tends to 0. Considerβ ={b₁, b₂, . . . , b_2m} an even subset of boundary half-edges of Ω. The set of configurations Conf_Ω(β) is defined as the set of all even degrees subgraphs of Ω containingβ and not containing any other boundary half-edges of Ω. It is easy to see that each configuration in Conf_Ω(β) consists of m paths linking boundary half-edges contained in β and possibly several loops. The weight of a configuration γ ∈ Conf_Ω(β) and the partition function are defined by

w(γ) =x^#edges(γ)n^#loops(γ), Z_Ω^β = X

γ∈ConfΩ(β)

w(γ),

where in #edges(γ) each half-edge contained inγ is counted as one half, parametersnand x are non-negative real numbers which can be chosen independently. We restrict our attention to n ∈ [0,2] and x = 1/p

2 +√

2−n which is conjectured to be the critical value [Nie82].

Two particularly interesting cases of the loop O(n) model aren= 0 andn= 1:

— If n = 0 and β = {b, b⁰}, then configurations of Conf_Ω(β) are self-avoiding walks from b to b⁰ in Ω because configurations with loops have zero weights. Main ques- tions for this model remain wide open, one of the biggest conjecture being conver- gence of the distribution on walks to SLE_8/3. The latter is known conditionally on the existence of the limiting measure and its conformal invariance [LSW04a]. Re-

1.1. LOOPO(N)MODEL. DEFINITIONS, CONJECTURES AND KNOWN RESULTS13 b⁰

b

Figure 1.1: Domain Ω and graph Ω_δ. All vertices Vδ(Ω) are marked by black and white squares, where the black colour corresponds to the boundary vertices ∂Vδ(Ω). Edges Eδ(Ω) are depicted by dotted and bold lines. Bold edges form a configuration of the loop O(n) model with Dobrushin boundary conditions (two half-edgesbandb⁰ on the boundary). These boundary conditions impose the existence of a unique path linkingbandb⁰. We call this path theinterface.

cently it was shown [DCS12b] that the connective constant of the self-avoiding walk on the hexagonal lattice isp

2 +√

2, thus confirming the critical value ofx forn= 0.

See [DCS12a,BDCGS11,MS93] for a survey of known results about the self-avoiding walks.

— If n = 1, then the weight of a configuration depends only on the number of edges in it. This case corresponds to the Ising model, where several important results on the conformal invariance were obtained recently. For details about the correspondence and known results about the Ising model see section 1.3.3and references therein.

Nienhuis [Nie84] related the loopO(n) model to the Coulomb gas, see [Mus10] for more details on the subject. This allowed him to predict the exact values of the universal critical exponents. Expressing the central charge corresponding to the loopO(n) model as a function ofn, one obtains

c= (6−κ)(3κ−8)/2κ, where κ= 4π/(2π−arccos(−n/2)). (1.1.1) In particular, this description would mean that for Dobrushin boundary conditions, i.e.β= {b, b⁰}, the interface frombtob⁰(see fig.1.1) converges to SLE_κin the scaling limit (see [Smi06, Smi10b] for more details).

The same conjecture can be obtained by considering the parafermionic observables intro- duced by Smirnov in the case of the FK cluster models in [Smi10a,Smi06]:

F_Ω(b, z;σ) = X

γ:b→z

x^#edges(γ)n^#loops(γ)e^{−iσwind(γ)}, (1.1.2) where the sum is taken over the configurations of the loop O(n) model containing a path from the boundary half-edgeb to a midpoint of an edgez, wind(γ) denotes the full rotation

of this path as one goes from b to z and σ is a parameter. One knows the complex phase of the parafermionic observable on the boundary and for σ= 1−3 arccos(−n/2)/4π the ob- servable partially satisfies the discrete Cauchy-Riemann equations. This led Smirnov [Smi06]

to conjecture that the parafermionic observable, when properly normalised, converges to the unique holomorphic solution of the corresponding boundary value problem in the continuous domain:

δ^−σ·F_Ω(b, z)/Z_Ω^b,b0 → C ·(ϕ⁰/ϕ)^σ, (1.1.3) where C is a lattice dependent constant and ϕ is a conformal map from Ω onto the upper half-plane. Note that the lefthand side in (1.1.3) is a martingale with respect to the growing interface and (ϕ⁰/ϕ)^σ is a martingale with respect to SLE_κ forκ= 3/(σ+ 1/2) (one needs to consider slit domains, see [Smi06]). Replacingσ by its expression in terms ofn, one gets the same value ofκ as the one derived via the Coulomb gas formalism.

The Conjecture (1.1.3) was proven for the Ising model, i.e. n = 1. The work in this direction was originated by Smirnov [Smi10a] who proved the same statement for the FK Ising model. A series of papers [CS11, CS12, DCHN11, KS12] by a group of authors led to [CDCH⁺14] establishing convergence of the interface to the SLE₃. For other values of n the conjecture remains open, a certain progress being achieved in the case of the self- avoiding walks, i.e. n = 0, where the connective constant was computed [DCS12b] using the parafermionic observable. Both the conjecture and a partial progress for certain models motivate the interest in other combinatorial observables satisfying local relations and having (conjectural) conformally covariant limits, especially in the case of the self-avoiding walks where the conformal invariance remains a big challenge for the probabilistic community.

1.2 Self-avoiding walk

Flory and Orr [Flo53,Orr47] introduced self-avoiding walk as a model of a long chain of molecules. Despite the simplicity of its definition, the model has proved resilient to rigorous analysis. While in dimensionsd≥5 lace expansion techniques provide a detailed understand- ing of the model, and the case d= 4 is the subject of extensive ongoing research, very little is known for dimensions two and three.

In Chapter4we use combinatorial techniques to prove two intuitive results for dimensions d≥2. We feel that the interest of this chapter lies not only in its results, but also in techniques employed in the proofs. To this end, certain tools are emphasised as they may be helpful in future works as well.

We mention two results from the early 1960s that stand at the base of our arguments:

Kesten’s pattern theorem, which concerns the local geometry of a typical self-avoiding walk, and Hammersley and Welsh’sunfolding argument, which gives a bound on the correction to the exponential growth rate in the number of such walks.

Let d≥ 2. For u ∈ R^d, let ||u|| denote the Euclidean norm of u. Let E(Z^d) denote the set of nearest-neighbour bonds of the integer lattice Z^d. A walk of length n ∈ N is a map γ : {0, . . . , n} → Z^d such that (γi, γi+1) ∈ E(Z^d) for each i ∈ {0, . . . , n−1}. An injective walk is calledself-avoiding. Let SAW_n denote the set of self-avoiding walks of length nthat start at 0. We denote by PSAWn the uniform law on SAW_n, and by ESAWn the associated expectation. The walk under the lawPSAWn will be denoted by Γ.

1.3. MAIN RESULTS 15 The law of the endpoint displacement under PSAWn is a natural object of study in an inquiry into the global geometry of self-avoiding walk. The displacement is quantified by the Flory exponentν, specified by the conjectural relationESAWn[||Γ_n||²] =n^2ν+o(1).

In dimensiond≥5, it is rigorously known thatν = 1/2 (see Hara and Slade [HS91,HS92]).

When d = 4, ν = 1/2 is also anticipated, though this case is more subtle from a rigor- ous standpoint. Recently, some impressive results have been achieved using a supersym- metric renormalisation group approach for continuous-time weakly self-avoiding walk: see [BDCGS11,BIS09,BS10] and references therein.

Whend= 2,ν = 3/4 was predicted nonrigorously in [Nie82,Nie84] using the Coulomb gas formalism, and then in [Dup89,Dup90] using Conformal Field Theory. It is also known subject to the assumption of existence of the scaling limit and its conformal invariance [LSW04a].

Unconditional rigorous statements concerning the global geometry of the model are almost absent in the low dimensional cases at present. In [DCH13], sub-ballistic behaviour of self- avoiding walk in all dimensionsd≥2 was proved, in a step towards the assertion thatν <1.

1.3 Main results

1.3.1 Self-avoiding walk on Z² with integrable weights

There are not so many rigorous statements about the self-avoiding walks in dimension 2, one of the main conjectures being convergence to SLE(8/3). Some progress in this direction was achieved by G. Lawler, O. Schramm and W. Werner, who proved in [LSW04a] that if the scaling-limit of self-avoiding walk exists and is conformally invariant, then it is SLE(8/3).

In 1984, B. Nienhuis nonrigorously derived in [Nie82] that the connective constant for the hexagonal lattice equals top

2 +√

2. This has been proved recently by H. Duminil-Copin and S. Smirnov in [DCS12b]. Since the self-avoiding walk on the square lattice does not seem to be integrable, it is not reasonable to expect any explicit formula for the connective constant in this case. Nevertheless, one can study natural variations of the model, for instance by introducing additional weights.

We fixθ∈_π

3,^2π₃

and consider the self-avoiding walk on Λ — the skewedZ² lattice with edges having length 1 and all plaquets having angles θ and π−θ.

To be precise this will be a curve starting and ending at the midpoints of edges, intersecting edges at right angles and having in each plaquet either one straight line connecting two opposite edges or two arcs surrounding opposite vertices or one arc or just no arcs (see fig. 3.1). Each rhombus has a weight according to the configuration of arcs inside it (see fig. 3.1):

— empty plaquet has weight 1,

— plaquet with an arc of angle θhas weightu₁,

— plaquet with an arc of angle π−θhas weightu₂,

— plaquet with a straight line has weightv,

— plaquet with two arcs of angleθ has weightw1,

— plaquet with two arcs of angleπ−θ has weightw₂.

The weight of the whole walk is calculated as the product of weights of the plaquets.

Denote one of the mid-edges of the lattice by 0. The partition function is equal to the sum

u1 u2 v w1 w2

1 θ

θ

u₁

u₁ u₁

u₂

u₁ w₁

u₁ v

v v v

Figure 1.2: Different ways of passing a rhombus with their weights and an example of a walk of weightu₁(θ)⁵u₂(θ)v(θ)⁴w₁(θ) and length 12.

of the weights of all self-avoiding walks on Λ starting at 0:

ω(γ) = Y

r−rhombus

ω(r), Z(u₁, u₂, v, w₁, w₂) =X

γ

ω(γ).

Let us consider a weighted number of self-avoiding walks of lengthndefined by

˜ c_n= 1

uⁿ₁ X

|γ|=n

ω(γ),

where by|γ|we mean the number of arcs inγ (straight passing of a rhombus counted as one arc). By definition, we find

Z(u₁, u₂, v, w₁, w₂) = X∞ n=0

˜ c_nuⁿ₁.

The next theorem computes the rate of of growth of the weighted number of self-avoiding walks.

Theorem 1. There exists a family of weights (u₁,u₂,v, w₁, w₂)_θ parametrised byθ∈[^π₃,^2π₃ ] such that for these weights lim_n→∞ √ⁿ

˜

c_n exists and is equal to _u¹

1. Furthermore, these weights can be calculated explicitly:

1.3. MAIN RESULTS 17

u₁ = sin(^5π₄ ) sin(^5π₈ +^3θ₈ )

sin(^5π₄ +^3θ₈) sin(^5π₈ −^3θ₈ ), (1.3.4) u₂ = sin(^5π₄ ) sin(^3θ₈ )

sin(^5π₄ +^3θ₈) sin(^5π₈ −^3θ₈ ), (1.3.5) v= sin(^5π₈ +^3θ₈ ) sin(−^3θ₈)

sin(^5π₄ +^3θ₈) sin(^5π₈ −^3θ₈ ), (1.3.6) w₁ = sin(^5π₈ +^3θ₈) sin(^5π₄ −^3θ₈ )

sin(^5π₄ +^3θ₈) sin(^5π₈ −^3θ₈ ), (1.3.7) w₂ = sin(^15π₈ +^3θ₈) sin(−^3θ₈)

sin(^5π₄ +^3θ₈) sin(^5π₈ −^3θ₈ ). (1.3.8) Theorem 2. Consider another way to define |γ|: a θ-arc has length 1, a (π−θ)−arc and a straight segment have any positive integer length (possibly different from each other). Then Theorem 1 remains true, i. e. the limit of √ⁿ

˜

c_n is equal to _u¹

1.

Remark 1.3.1. The case θ = ^π₃ corresponds to the honeycomb lattice and there is a way to define|γ|in such a way that Theorem 2computes the connective constant of the honeycomb lattice (see Section 2.1).

The weights (1.3.4)-(1.3.8) were discovered by B. Nienhuis [Nie90] in 1990 as solutions of the Yang-Baxter equation. They were rediscovered by J. Cardy and Y. Ikhlef [IC09] in 2009 as the weights for which the parafermionic observable satisfies some particular equations. For the connection between these two approaches, see [AB14,IWWZJ13]. See also [dGLR13] for the weights on the boundary. We should just mention here that in [Nie90] and [IC09] a more general case is considered — the O(n) model with n ∈ [−2,2] (the self-avoiding walk is a particular case of this model for n = 0). Unfortunately, the weights written there contain some minor misprints, so for completeness we include a correct version of the weights in Section 2.5.

In the case θ= ^π₂ the weights are symmetric, i. e. u1 =u2 and w1 =w2. One can view a walk as a self-avoiding walk onZ² which is allowed to touch itself but each time gets penalised byw₁/u²₁ ≈0.675 and that gets penalised by v/u₁ ≈0.785 for each vertex it passes without a turn. Theorem1 confirms the conjecture [Bat] that the asymptotic of √ⁿ

˜

cn is equal to 1

u₁(π/2) = r

3 +1 2

q

26 + 7√

2 = 2.448. . .

This is below the predicted [GE88] value≈2.638 for a connective constant ofZ².

Another interesting question is the value of the critical fugacities for walks in a half-plane interacting with the boundary. For the self-avoiding walk in the half-plane insertion of a fu- gacity means favouring each additional visit of the border. One can define the critical fugacity such that self-avoiding walk with greater fugacities stick to the border. In [BBMdG⁺14] it was proven that the critical fugacity for the self-avoiding on the hexagonal lattice is equal to 1 +√

2. It would be natural to generalise this computation to the case of the self-avoiding walk on Z² with integrable weights given by (1.3.4)-(1.3.8). Though we conjecture that a

a b

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

0

Figure 1.3: On both pictures we depict the Θ-tiling of the right half-plane, where Θ is a sequence of angles θ₁, θ₂, etc. Left: a path contributing to the 2-point funtion G_Θ(a, b).

Right: bridges contributing toB_Θ,6.

similar statement should hold, it is unclear how to formulate it as there are various ways to touch the boundary in this case. Instead of that we show that the 2-point function (i.e. the sum of weights of all walks with two fixed endpoints) for the half-plane does not depend on the choice of a rhombic tiling. In order to see how these two questions are related, note that there is a canonical decomposition of a self-avoiding walk touching the boundary into several arcs and a path which does not touch the boundary. Hence, the critical fugacity describes the competition between the 2-point function and the total weight of walks which never revisit the boundary line.

Pick two pointsaandb with integer coordinates on the boundary of the right half-plane.

Consider a series of angles θ₁, θ₂, θ₃, . . ., where each angle is between π/3 and 2π/3, and denote it by Θ. ByG_Θ(a, b) we denote the sum of weights of all walks in the right half-plane tiled with unit rhombuses of angleθ₁ in the first column, with angleθ₂ in the second column, etc (see fig.1.3). Such partition function is usually called the 2-point function. ByG_π/3(a, b) we denote the 2-point function for the hexagonal lattice with edges of length 1/√

3. We need to show thatG_Θ(a, b) =G_π/3(a, b).

Theorem 3. Let Θ be any series of anglesθ₁, θ₂, θ₃, . . . between π/3 and2π/3. Then in the right half-planeGΘ(a, b) =G_π/3(a, b) for any two points aand b on the boundary.

Remark 1.3.2. One can considerθ < ^π₃ (orθ > ^2π₃ ) but the weightw2 (orw1) becomes nega- tive, so we do not address this question here. Given that all angles are strictly betweenπ/3 and 2π/3, there is a unique way to tile the half-plane (i.e. the one described in the theorem above). If some angles are equal toπ/3 or 2π/3 and the rest are betweenπ/3 and 2π/3, then one could also have several strips containing vertical rhombuses of angleπ/3. As noted above, in the caseθ =π/3 (or 2π/3) a rhombus can be split into two equilateral triangles and this corresponds to the classical self-avoiding walk on the hexagonal lattice independently of the initial rhombic tiling.

For the self-avoiding walk on the hexagonal lattice the limit of the total weight of bridges in a strip of a fixed width was shown to be equal to 0 (Theorem 10 in [BBMdG⁺14]). We

1.3. MAIN RESULTS 19 give a new short proof of this statement and using Theorem3 deduce that this limit equals to 0 for the self-avoiding walk on any rhombic tiling with integrable weights. In the notation of theorem3, we introduce the partition function of bridges of width T:

B_Θ,T = X

z∈k-th column

X

γ: 0→zbridge

w(γ),

where the first sum is taken over all points on the right border of the k-th column of the Θ- tiling of the right half-plane, and the second sum is running over all bridges linking 0 and z (see fig. 1.3).

Theorem 4. Let Θbe a series of angles θ₁, θ₂, θ₃,· · · ∈[π/3,2π/3]. Then B_Θ,T −−−−→

T→∞ 0.

1.3.2 Infinitesimal deformations of the discrete complex structure and the discrete stress-energy tensor

In Chapter3 we suggest a new combinatorial observable obtained by evaluating the par- tition function after a local deformation of the lattice along the integrable line. To the best of our knowledge there is no known integrable way to define the loop O(n) model on a de- formed hexagonal lattice by adding non-homogeneous weights on the edges. On the other hand, one can switch to the dual lattice and view the loop O(n) model as a model on the plane tiled with equilateral triangles. Gluing several pairs of adjacent triangles, we get a model on the plane tiled with lozenges with anglesπ/3 and 2π/3. In each non-empty lozenge a local configuration is one of 4 types, and a weight of a lozenge depends only on the total length of arcs traversing it (see fig.3.2). Adding one more possible local configuration, Nien- huis discovered [Nie90] a family of integrable weights of lozenges satisfying the Yang-Baxter equation, the weights being parametrized by the angle of a lozenge. Later it was pointed out by Cardy and Ikhlef [IC09] that for these weights the parafermionic observable still satisfies (as it does on the hexagonal lattice) a part of the Cauchy-Riemann equations; see [AB14]

for a connection between the two approaches. This brings us to a definition of an integrable loop O(n) model on a Riemann surface tiled by rhombi and equilateral triangles and allows to deform each rhombus (see section3.1for details).

In particular, we will be interested in small perturbations of the original triangular tiling dual to the hexagonal lattice. Two examples that we are going to consider are: replacing two adjacent triangles by a rhombus with anglesθ and π−θ forθ close to π/3; inserting a rhombus with angles θ and π−θ for a small θ. First deformation corresponds to the edges of the hexagonal lattice and second is defined for each midline of a hexagon (this is the place where an infinitesimal rhombus is inserted). Both times we calculate the partition function on a new lattice and get Z_Ω^β(e, θ) for an edge e and Z_Ω^β(m, θ) for a midline m of a certain hexagon. The real-valued observables are defined as derivatives of these partition functions:

T_edge^β (e) :=c_edge+ d

dθlogZ_Ω^β(e, θ)

θ=^π₃ T_mid^β (m) :=c_mid+ d

dθlogZ_Ω^β(m, θ)

θ=0, (1.3.9) where constants c_edge, c_mid come from the infinite-volume limit of the model. For n = 1 these constants can be computed explicitly, see eq. (1.3.10). Note that, alternatively, one can give a combinatorial definition of both observables, staying with the loop O(n) model on the hexagonal lattice and considering sums over the sets Conf^[e]_Ω(β) and Conf^[m]_Ω (β) of

configurations with defects oneorm, see fig.3.4and precise definition in Section3.1.3. From this definition it is easy to see that for Dobrushin boundary conditions Tedge and Tmid are martingales with respect to the growing interface.

Remark 1.3.3. In our definition of the loop O(n) model on a deformed lattice we keep the loop-weight equal ton and change only the edge-weightx. Note that this is consistent with standard considerations of the loop O(n) model on the cylinder, e.g. see [Car13, Car96].

It is probably necessary to deform the loop-weight to understand better the relation with the Coulomb gas following the idea of Nienhuis [Nie82] of describing the full weight of a configuration as product of local weights. This is another interesting question and we do not address it here. We note that in this case the derivative of the partition function will get an additional term corresponding to varying the loop-weight. This term is closely related to the height function.

The real-valued observables T_edge^β , T_mid^β describe the respond of (the partition function of) the loopO(n) model in a given discrete domain to an infinitesimal change of the discrete complex structure. We construct a complex-valued observableT^βwhich projections on certain directions are given byT_edge^β andT_mid^β . In field theories the corresponding operator is called the stress-energy tensor. Hence, we refer to the observablesT_edge^β ,T_mid^β as a discrete stress-energy tensor. According to the conformal field theory, the stress-energy tensor has the Schwarzian conformal covariance:

hT(w)O₁(ϕ(z₁)). . . O_k(ϕ(z_k))iΩ =

(ϕ⁰(w))²hT(ϕ(w))O₁(ϕ(z₁)). . . O_k(ϕ(z_k))iΩ⁰

+₁₂^c[Sϕ](w)hO₁(ϕ(z₁)). . . O_k(ϕ(z_k))iΩ⁰

k

Y

j=1

|ϕ⁰(z_j)|^∆j,

where [Sϕ](w) := ^ϕ_ϕ⁰⁰⁰_0(w)^(w) − ³₂h

ϕ⁰⁰(w) ϕ⁰(w)

i2

is the Schwarzian derivative of a conformal mapping ϕ : Ω→Ω⁰, andO_j(z_j) stands for a real-valued primary field atz_j ∈Ω\ {w} with a scaling exponent ∆j, and c is the central charge of the theory.

As we already mentioned above, the loopO(n) model is conjectured to have a conformally invariant scaling limit which can be described by the CFT with a central charge given by eq. (1.1.1). Hence, we conjecture that T^b,b0, when properly normalized, converges to the unique martingale of SLE_κ which has the Schwarzian conformal covariance.

Conjecture T.There exists a lattice dependent constantC such that forσ= 3/κ−1/2 δ⁻²T^∅ → C · c

12[Sϕ](w), δ⁻²T^b,b0 → C · σ

ϕ⁰(w) ϕ(w)

2

+ c

12[Sϕ](w)

! .

The functionT^b,b0 is a martingale with respect to the growing interface frombtob⁰. Hence, the proof of the above conjecture would imply convergence of the interface to the SLE_κ. This is similar to the approach of the parafermionic observables proposed in [Smi06]. As a strong support for this conjecture, we prove it for n = 1 (Ising model), see section 1.3.3.

For everyn∈ [0,2], components of the discrete stress-energy tensor T satisfy certain linear relations which can be interpreted as a part of the discrete Cauchy-Riemann relations (see section3.1).

1.3. MAIN RESULTS 21 We note that on the boundary the conjecture forn= 0 agrees with the restriction property of the SLE_8/3. Indeed, in this case one has c = 0 and Conjecture T means that T^b,b0, after a proper rescaling, converges to ⁵₈|ϕ⁰/ϕ|²·δ². On the boundary the latter is the probability to intersect theδ-neighbourhood of a boundary point. The boundary values of T^b,b0 can be interpreted in a similar way. Indeed, by definition of T^b,b0 given in section 3.1.3, only paths touching a boundary edge contribute to the value of T^b,b0 at it.

1.3.3 Convergence results for the Ising model

Let us now discuss the special case n = 1, which corresponds to the Ising model with spins assigned to the faces of a discrete domain Ω. This correspondence works as follows:

given a configurationγ ∈Conf_Ω(β) with the boundary conditionsβ={b₁, . . . , b_2m}, one puts spins±1 on faces of Ω (including boundary ones) so that two spins at adjacent faces are differ- ent if and only if their common edge belongs to γ, with boundary spins being +1 everywhere except along the counterclockwise boundary arcs (b_2k−1b_2k) (where, on the contrary, they are fixed to be −1). Below we also use the notation ‘+’ instead of ∅ for the empty boundary conditions in order to emphasize this choice of the sign of boundary spins. It is easy to see that the partition functions of the loop O(1) model discussed above can be written as

Z_Ω^β = P

σ∈Conf^spin_Ω (β)x^#_crit^{{u∼w:σu6=σw}−12|β|}

= x

1

2(|E(Ω)|−|β|) crit

P

σ∈Conf^spin_Ω (β)exp₁

2logx_crit·P

u∼wσ_uσ_w , wherex_crit= 1/√

3, we use the notation Conf^spin_Ω (β) for the set of all possible spin configura- tions on the faces of Ω with boundary conditionsβ, and #{u∼w: σ_u 6=σ_w} is the number of pairs of adjacent faces of Ω carrying opposite spins. The value x_crit = 1/√

3 is known to be critical for the Ising model on faces of the regular honeycomb lattice, see [Wan50].

It is well-known in the theoretical physics context (e.g., see [DFMS97] or [Mus10]) that the limit of the critical Ising model asδ→0 can be described by the Conformal Field Theory (of a Free Fermionic Field) with the central chargec= ¹₂. Before 2000s, this passage was usually considered in the infinite-volume (or half-infinite-volume) setup. During the last decade new techniques appeared, which led to a number of rigorous convergence results (confirming the existing CFT predictions) for various correlators in the critical Ising model considered on discrete approximations Ω_δ to ageneral planar domain Ω. The first results of this kind were obtained by Smirnov in [Smi06,Smi10a] for holomorphic observables in the so-called FK-Ising model (also known as the random cluster representation of the Ising model). In a similar manner, one can use the combinatorial definition (1.1.2) in the case n = 1 (and σ = 1/2) to view these observables as solutions to some well-posed discrete Riemann-type boundary value problems in Ω_δ. This opens a way for the treatment of their limits asδ→0 in general planar domains Ω. Further developments of the techniques proposed in [Smi06, Smi10a]

include the convergence results for the basic fermionic observables [CS12], energy density correlations [HS13, Hon10] and spin correlations [CI13, CHI15], the latter relying upon the convergence of discrete spinor observables, see Section3.2.4 below for their definition.

For the Ising model we use the following notation for the limits from the Conjecture T (substitutingc= 1/2):

hT(w)i⁺_Ω := [Sϕ](w)

24 , hT(w)i^b,b_Ω⁰ := 1 2

ϕ⁰(w) ϕ(w)

2

+ [Sϕ](w) 24 ,

whereϕ: Ω→His an arbitrary uniformization mapping in the first case and any one of those sending {b, b⁰} to{0,∞} in the second. Moreover, in this case the constants c_edge and c_mid in the definition (1.3.9) of Tedge and Tmid can be computed explicitly and take the following values:

c_edge=−_2π¹ + ¹

4√

3, c_mid=−_π¹ + ¹

3√

3. (1.3.10)

We now formulate the first of our convergence results. For technical reasons, in the case of Dobrushin boundary conditions, we assume that the boundary points lie on straight parts of the boundary of Ω which are orthogonal to the edges of the lattice approximation, see also remark1.3.4.

Theorem 5. Let Ω⊂ C be a bounded simply connected domain, w ∈ Ω and b, b⁰ ∈∂Ω two points on the boundary. Let Ω_δ be a discrete approximation to Ω on the honeycomb grid Cδ

with mesh size δ, the boundary (half-)edges b_δ and b⁰_δ approximate b and b⁰, and w_δ be the face of Ω_δ that contains w. Then, for any fixed direction τ ∈ {1, e^2πi3 , e^4πi3 } of edges of Cδ, one has

δ⁻²T_edge⁺ (e_δ) ⇒ _π³Re[τ²hT(w)i⁺_Ω], δ⁻²T_mid⁺ (m_δ) ⇒ _π³Re[(iτ)²hT(w)i⁺_Ω], δ⁻²T_edge^bδ,b0δ(e_δ) ⇒ _π³ Re[τ²hT(w)i^b,b_Ω⁰], δ⁻²T_mid^bδ,b0δ(m_δ) ⇒ ³_πRe[(iτ)²hT(w)i^b,b_Ω⁰], as δ→0, wheree_δ is any of the two edges ofw_δ oriented in the direction τ,m_δ is the midline of w_δ orthogonal to e_δ, and the convergence is uniform on compact subsets of Ω.

Remark 1.3.4. The assumption about the orthogonality of the half-edges b_δ and b⁰_δ to the straight parts of the boundary ∂Ω can be relaxed using the results from [CS12]. Moreover, the techniques developed in [HK13] allow to relax these technical assumptions even further and prove a similar statement in the case of rough boundaries.

Remark 1.3.5. We use the following strategy to prove Theorem 5. It is not hard to see from the combinatorial definition of T_mid^∅ (m) that these quantities can be expressed via the values of discrete fermionic observables defined by (1.1.2), where n = 1 and σ = 1/2, see Section 3.2.2 for more details. More generally, for boundary conditions β with |β| = 2m one can exploit the fermionic nature of these observables and show that T_mid^β (m) admits an expression as a Pfaffian of some 2(m+ 1)×2(m+ 1) matrix with entries given by the values of the fermionic observables; see Section 3.2.3, where the case of Dobrushin boundary conditions β = {b, b⁰} is treated. Therefore, one can use existing discrete complex analysis techniques to derive Theorem5from the (known in the casen= 1) convergence results for the fermionic observables. In continuum, the fact that the stress-energy tensor T =−¹₂ :ψ∂ψ: can be expressed via fermions is a particular feature of the corresponding CFT, and there is no surprise that the similar discrete quantities converge to their putative limits asδ →0.

Nevertheless, it is worth noting that some additional terms appear when working with the geometrical definition given by (1.3.9). These additional terms have bigger scaling exponents and thus cancel out in the limit δ → 0, see Remark 3.2.2 and Proposition 3.2.3 for more details.

Let us now discuss the limits of correlation functions of the two discrete stress-energy tensors considered at distinct points of Ω. Though these quantities can be introduced in the general loopO(n) model context by considering several spatially separated infinitesimal