Holomorphic spinor observables and interfaces in the critical ising model

Thesis

Reference

Holomorphic spinor observables and interfaces in the critical ising model

IZYUROV, Konstantin

Abstract

We generalize Smirnov's discrete holomorphic observables in the critical Ising model to the case of multiply connected domains. Our observables are spinors, that is, they are multiplicatively multi-valued with monodromy -1. We prove their convergence to conformally covariant scaling limits as the mesh size tends to zero. As applications, we get partial results towards the proof of conformal invariance of the spin correlations, and develop a fairly general theory of scaling limits of multiple Ising interfaces in multiply connected domains.

IZYUROV, Konstantin. Holomorphic spinor observables and interfaces in the critical ising model. Thèse de doctorat : Univ. Genève, 2011, no. Sc. 4394

URN : urn:nbn:ch:unige-184246

DOI : 10.13097/archive-ouverte/unige:18424

Available at:

http://archive-ouverte.unige.ch/unige:18424

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

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

Section de Math´ematiques Professeur Stanislav Smirnov

Holomorphic Spinor Observables and Interfaces in the Critical Ising 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

Konstantin Izyurov

de

Saint-Petersburg (Russie)

Th`ese N^◦ 4394

Gen`eve

Atelier d’impression ReproMail de l’Universit´e de Gen`eve 2012

Abstract

The thesis is devoted to the study of the critical Ising model in two dimensions.

Its main contents is based on two papers: “Holomorphic spinor observables in the critical Ising model” and “On the critical Ising interfaces in multiply-connected domains” (the former one is joint with Dmitry Chelkak). The results consist in generalizing the holomorphic observable of Smirnov, and deriving some applications of this generalization.

In the case of multiply-connected domain Ω approximated by a discrete domain Ω^δ, we construct a discrete holomorphic spinor observableF$for every double cover

$ of Ω^δ. This observable changes sign when one changes the sheet of the double cover. We show that the observables satisfy certain boundary conditions, and prove their convergence to continuum solutions of corresponding boundary value problems as the mesh size tends to zero. As an application, we show that it is possible to express certain ratios of spin correlation in terms of those observables, thus proving convergence of these ratios to conformally invariant limits. This confirms predictions made earlier by physicists, and, furthermore, provides new explicit formulas.

The second part of the work is concerned with the Ising model interfaces. We consider a fairly general setup of finitely-connected domain with multiple boundary change operators, and thus multiple discrete curves connecting the points on the boundary of the domain. Considering one of these multiple interfaces, we describe it by Loewner evolution. We show that, as the mesh size tends to zero, the driving force converges to a (random) function at which satisfies the stochastic differential equation da_t = √

3dB_t+ 3∂_alogZ. Here Z is the scaling limit of the Ising model partition function in the domain g_t(Ω), with corresponding boundary conditions (namely, boundary change operators at at and the images of the marked points underg_t), and g_t is the Loewner map. This scaling limit can be described in terms of scaling limits of the observables considered in the first part of the thesis, and thus in terms of the (unique) holomorphic solution to a boundary value problem.

Depending on the configuration of marked points, the spinor observable with respect to appropriate double cover should be used. In the case of simply-connected and doubly-connected domains, the explicit expressions for drifts are available. This leads, in particular, to a rigorous proof of a four-point crossing formula for the Ising model. We also consider the case of doubly-connected domain with a single interface connecting the outer boundary component to the inner one. We show that if the inner one shrinks to a point, the interface converges to radial SLE₃. To this end, we use the reversibility of corresponding annulus SLE’s.

R´esum´e

Cette th`ese est d´evolue au mod`ele d’Ising critique en deux dimensions. Son contenu principal est bas´e sur les articles “Holomorphic spinor observables in the critical Ising model” (avec Dmitry Chelkak), et “On the critical Ising interfaces in multiply-connected domains”. Nous g´en´eralisons l’observable holomorphe discr`ete de Smirnov, et d´erivons plusieurs applications de cette g´en´eralisation.

Dans le cas d’un domaine multiplement connexe Ω approxim´e par un domain discr`ete Ω<sup>δ</sup>, nous construisons une observable holomorphe discr`eteF_$ pour chaque revˆetement double $ de Ω^δ, qui change signe quand on passe `a l’autre feuille du revˆetement. Nous d´emontrons que ces observables satisfont certaines conditions au bord, et ainsi prouvons leur convergence vers les solutions continues des probl`emes au bord correspondants quand la taille du r´eseaux tend vers z´ero. Comme appli- cation, nous montrons qu’on peut exprimer certains rapports des corr´elations des spins par notre observables. Ca donne convergence de ces rapports vers une limite conform´ement invariante. Ceci confirme quelques pr´edictions faites pr´ec´edemment par des physiciens, mais aussi donne de nouvelle formules explicites.

La deuxi`eme partie de la th`ese traite des interfaces dans le mod`ele d’Ising. Nous consid´erons une situation assez g´en´erale d’un domaine multiplement connexe avec plusieurs points de changement des conditions au bord, et donc plusieurs courbes discr`etes qui connectent les points sur le bord du domaine. En consid`erant une des ces interfaces, nous la decrivons par l’´evolution de Loewner. Nous d´emontrons que, quand la taille d’´echelle tend vers z´ero, le processus de guidage converge vers une fonction al´eatoire a_t qui satisfait l’´equation differentielle da_t = √

3dB_t+ 3∂_alogZ, ou Z est la limite d’´echelle de la fonction de partition du mod´ele d’Ising dans g_t(Ω) avec les conditions au bord correspondantes, et gt est l’application de Loewner.

Ces limites d’´echelle peuvent ˆetre exprim´ees comme les limites des observables con- sider´ees dans la premi`ere partie de la th`ese, et donc comme les solutions uniques d’un certain probl`eme au bord. Dependant de la configuration des points marqu´es, l’observable spinor par rapports au revˆetement bien choisi est employ´ee. Pour les domaines simplement connexes et doublement connexes, les expressions explicites pour le processus de guidage sont disponibles. Nous consid´erons aussi le cas d’un domaine doublement connexe, avec une seule interface qui joint le bord ext´erieur au bord int´erieur. Nous prouvons que si `a la limite, ce derni`ere tend vers un point, l’interface converge vers le SLE3 radial. A cet effet, nous utilisons la reversibilit´e du SLE radial.

Contents

1 Introduction 7

1.1 Ising model background . . . 7

1.2 The results of the thesis . . . 8

1.2.1 Holomorphic spinor observables . . . 8

1.2.2 Interfaces and SLE3 processes . . . 10

1.3 Organization of the thesis . . . 12

1.4 Acknowledgments . . . 13

2 Spinor observables and their discrete holomorphicity 15 2.1 Notation and conventions . . . 15

2.2 Basic observables . . . 16

2.3 Multi-source observables and pfaffian formulae . . . 19

2.4 Relation to partition functions and spin correlations . . . 21

3 Convergence results for observables 25 3.1 Statements of the theorems . . . 25

3.2 Proof of convergence theorems for the observables . . . 27

3.3 Convergence and properties of multi-source observables . . . 37

3.4 Application: scaling limits of ratios of spin correlations . . . 40

3.5 Explicit computations in the half-plane . . . 42

3.6 Relation to the energy density and spin fields . . . 44

4 Interfaces and their scaling limits 46 4.1 Discrete martingale property . . . 46

4.2 Determination of driving processes: the general formula . . . 48

4.3 Determination of driving processes: explicit examples . . . 50

4.3.1 Four marked points in a simply-connected domain . . . 50

4.3.2 Annulus computations . . . 51

5 The proof of convergence of the interfaces 55 5.1 Setup and some definitions . . . 55

5.2 Main theorem and outline of the proof . . . 56

5.3 Proof of technical lemmas . . . 59

5.3.1 Compactness and uniform convergence of the observables . . 59

5.3.2 Analytic properties of continuous observables . . . 61

5.3.3 Deriving the equations (5.6) . . . 65

5.3.4 Existence and uniqueness . . . 67

5.4 Some remarks . . . 70

5.4.1 Absolute continuity and Girsanov transform . . . 70

5.4.2 On the proof of crossing formula (4.8) and convergence of in- terfaces as paths . . . 71

5.4.3 Radial SLE3 case . . . 73

1 Introduction

1.1 Ising model background

In his celebrated doctoral thesis on magnetism, Pierre Curie, in particularly, dis- covered a phase transition in ferromagnets or ferrimagnets. For each ferromagnetic (or ferrimagnetic) material, there is a critical temperatureT_c (now called the Curie point) above which it becomes paramagnetic. For iron, cobalt and nickel, the Curie temperature is equal to 1143K, 1303K and 631 K respectively.

In 1920, Wilhelm Lenz proposed a statistical mechanics model which aims to explain this change of behavior. In this model, each site in the crystal lattice is assumed to carry a magnetic momentum, or spin. The magnetic forces try to align neighboring spins in the same direction, while the heat motion brings a disorder in the configuration. The phase transition was supposed to be explained by the competition between these two effects.

The configurations in the Ising model are functionsσdefined on the set of lattice sites and taking values in the set {1,−1} (that is, configurations are all possible assignments of spins to the lattice sites). The probability of a configuration σ is proportional to

exp[1 T

X

x∼y

σ(x)σ(y)],

where the sum is taken over all pairs of neighboring sites of the lattice, andT is the temperature. Thus the model favors the configurations with aligned spins, but this effect decreases as the temperature grows. AtT =∞, all configurations are equally probable, thus the model is completely disordered, while in the limit T → 0, only the two completely aligned configurations survive.

In his doctoral thesis of 1925 [Isi25], Ernst Ising solved the model in one dimen- sion, and showed that it obeys paramagnetic behavior at any positive temperature.

He conjectured this result to be also true for upper dimensions, which was found to be wrong in the 1936 work by Peierls [Pei36], who proved the existence of phase transitions in dimension greater or equal to two. Another important milestones in the study of the Ising model are Kramers-Wannier work of 1941 [KW41], introduc- ing a high-temperature expansion and discovering duality in the Ising model, thus identifying the critical point as the self-dual one, and the work of 1944 by Onsager [Ons44], who computed the free energy and derived critical exponents.

There was a gradual understanding in the physics community that many statis- tical models in two dimensions, and in particular the Ising model, should exhibit remarkable symmetries at criticality. This finally led to a conjecture that critical Ising model is conformally invariant. The conformal invariance means that observ- ables in the model, such as spin or energy correlations, have conformally invariant or covariant scaling limits when the mesh size goes to zero. Moreover, it was con- jectured that these limits should be described by one of the conformal field theories introduced in [BPZ84]. The correlations in these CFT’s are amenable to explicit computations.

The proofs of there conjectures were out of reach until recently. In 2000, Schramm [Sch00] proposed his celebrated tool to study the conformally invariant 2D lattice model, namely, the Stochastic Loewner Evolutions (SLE). These are random pla- nar curves that may be thought of as continuous counterparts of the interfaces, or discrete curves, in lattice models. In the case of the Ising model, given a spin con- figuration, one can consider the subset of edges that separate pluses from minuses.

These edges form loops, and even when the mesh size goes to zero, some of these loops still have macroscopic size with high probability. With an appropriate choice of boundary conditions, one can obtain a distinguished interface connecting two boundary points. As the mesh size goes to zero, this (random) interface converges to a random continuous curve SLE3, and the whole loop collection is conjectured to converge to random continuous loop ensemble CLE₃.

Although convergence of interfaces to SLEκ is conjectured in many lattice mod- els at criticality, only a few of those conjectures are rigorously proven so far. These include loop-erased random walks and uniform spanning trees (SLE₂and SLE₈corre- spondingly, [LSW04]), critical percolation (SLE6, [Smi01]) and external perimeters of Brownian motions (SLE⁸

3, [LSW03]), harmonic explorers and level lines of the Gaussian Free Field (SLE₄, [SS05, SS09]). We refer the reader to [Law05, Wer04]

for the background on Schramm-Loewner evolution.

In 2006, Smirnov proved conformal invariance of fermionic observables in the Ising model [Smi06]. This, in particular, led to the proof of convergence of the Ising and FK-Ising interfaces to SLE3 and SLE¹⁶

3 curves respectively [KS09, CS11]. In the latter case a description of the full scaling limit of loop soups in terms of the SLE(¹⁶₃ ,−²₃) was obtained. Chelkak and Smirnov [CS09] later proved universality of the critical Ising model on isoradial graphs.

The proof of conformal invariance of fermionic observables was highly motivated by the emerging theory of SLE curves. However, applications of this result reach far beyond SLE questions. Using slightly different version of the observable, Hongler and Smirnov were able to compute the scaling limit of the energy density [HS10].

This result was later extended to the proof of conformal invariance of all energy correlations, and certain boundary spin correlations [Hon10]. In the FK-case, the fermionic observables were also applied to prove RSW-type bounds for FK-Ising model.

Despite the above results, the program of full and rigorous understanding of conformal invariance in the Ising model is still not completed. In the CFT part of the story, the existence and conformal invariance of the scaling limit of the bulk spin field (and its correlations with other fields) is out of reach for the moment. The case of multiply connected domains and Riemann surfaces is also much less studied. The interfaces in a more general setup as just one “chordal” interface, such as multiple interfaces, or radial-type interfaces, or interfaces in multiply-connected domains are also worth study. All these questions are of particular interest because the Ising model is well suited to the study of conformally invariance, and therefore one may hope to obtain eventually a fairly complete and rigorous picture. The present work pursues these directions.

1.2 The results of the thesis

1.2.1 Holomorphic spinor observables

The first part of the thesis is based on the joint article with Dmitry Chelkak [ChI11].

The main result of that part is the generalization of holomorphic observables and extension of convergence results to the case of a multiply-connected domain Ω. We show that in this case, along with the well-known observableF0proposed by Smirnov in [Smi06], one can define a number of new observables F$, one for each double cover $ of Ω (for k-connected domains there exist 2^k−1 such double covers), see Definition 2.1. These new observables are proven to be discrete holomorphic and

satisfy the equationF$(z) =−F_$(z^∗), ifz6=z^∗ belong to a fiber of the same point.

Hence, we call them holomorphic spinors. Note that F$ are closely related to the vector bundle technique developed by Kenyon in [Ken10] in the dimer model context and probably can be rewritten in that context via Fisher’s dimer representation of the Ising model.

We relate the values ofF_$at the boundary∂Ω^δto certain partition functions and spin expectations in the Ising model (see Proposition 2.12) and prove convergence of these observables to conformally covariant limits (see Theorem 3.2 and Section 3.2).

This leads us to a rigorous proof of a number of formulae that appeared in [BG93]

and earlier papers. Thus, if γ₁^δ, . . . , γ_k^δ denote inner components of ∂Ω^δ and a, bare two points on the outer boundary of Ω, then, for all m ≤k, we prove convergence of the ratios (see Corollary 3.24)

E_a^δ_b^δ[σ(γ₁^δ). . . σ(γ_m^δ) ] E+[σ(γ₁^δ). . . σ(γ_m^δ) ] →

δ→0 ϑ^(Ω)_ab (γ1, . . . , γm), (1.1) to conformally invariant limitsϑ, whereEa^δb^δ is the expectation for the Ising model with Dobrushin boundary conditions on the outer boundary (“−” on the (a^δb^δ) boundary arc and “+” on (b^δa^δ)) and monochromatic (i.e., constant but unknown) on inner components γ_j, E+ stands for the same expectation with all “+” outer boundary, and σ(γ_j) denote the spins of inner boundary components.

Note that we do not impose any assumptions concerning the size of γj. In particular, one can consider Ω^δ obtained from a simply connected domain Ω^δ₀ by deleting several single faces z_j^δ ∈Ω^δ₀. In this case (1.1) provides us the limit of the ratios of spin correlations in Ω^δ₀ taken at z₁, . . . , z_m. For instance, in the case of only one spin (magnetization) in a simply-connected Ω^δ₀, this limit is equal to

ϑ^Ω_ab^0\{z}(z) = cos [πhm_Ω0(z,(ab))],

where hm stands for the harmonic measure of the boundary arc (ab) fromz ∈Ω₀. Moreover, in Section 3.5 we give explicit formulae allowing one to compute all the limits of the ratios of spin correlations

E_a^δ_b^δ[σ(z₁^δ). . . σ(z^δ_m) ] E+[σ(z₁^δ). . . σ(z_m^δ) ] →

δ→0 ϑ(φ(z1), . . . , φ(zm)), (1.2) where ϑ = ϑ^C_∞,0⁺ are explicit functions and φ : Ω₀ → C+ is a conformal map from Ω0 onto the upper half-plane C+ sending a to ∞ and b to 0. To the best of our knowledge, apart from the case of one and two spins, these formulae are new even for the the physics literature. It is worthwhile to note that our proof also ensures that the same limit (1.2) appears for nontrivial inner boundary componentsγ_j^δshrinking to pointsz_j asδ tends to zero.

Further, closely following the route proposed by Hongler in [Hon10], we prove a pfaffian formula which generalizes (1.1) to the case of 2nboundary change operators (in other words, “+/ −/ +/−” boundary conditions with 2n marked boundary points, see Subsection 2.3 and 3.4). For m = 1,2 this pfaffian formula (along with the expressions forϑ) was previously derived by means of Conformal Field Theory [BG93], whereas we give a rigorous proof for generalmboth in discrete, and, thanks to the convergence theorem, in continuous settings.

The convergence result (1.2) and its generalization to the case of more general boundary conditions may be viewed as a step towards a far more ambitious program

of computing the scaling limit of the spin field. This is an ongoing project of the author together with Dmitry Chelkak and Cl´ement Hongler. In Subsection 3.6 we briefly explain how the spinor holomorphic observables can be exploited to attack this problem.

1.2.2 Interfaces and SLE3 processes

The simplest setup to study the interfaces in the Ising model is the case of simply- connected discrete domain Ω^δ with two marked points a^δ, b^δ on the boundary. One introduces the Dobrushin boundary conditions: “+” on the boundary arc (a^δb^δ) and

“−” on (b^δa^δ). This forces existence of the interface betweena^δ and b^δ separating pluses from minuses. If the mesh size tends to zero, and the domain (Ω^δ, a^δ, b^δ) approximates (in a suitable sense) a continuous domain Ω, a, b, then it was shown [CS11] that the interface converges to chordal SLE3 in Ω fromatob.

The SLE-type growth processes were also studied beyond the simplest case of chordal or radial SLE in simply-connected domains. There are several difficulties that arise in that case. In the case of simply-connected domains with two marked points, Schramm’s SLE curves can be characterized by Schramm’s principle: they are the only processes satisfying conformal invariance and domain Markov prop- erty. When the curve is described by Loewner evolution, these properties boil down to the conditions for the driving force to have independent, identically distributed increments, from which one concludes that it must be a multiple of the Brown- ian motion. Here it is crucial that all simply-connected domains with two marked points are conformally equivalent to each other. In the case of multiply-connected domains and/or multiple marked points, this is no longer the case, and thus the above argument fails.

Another difficulty, more of a technical nature, is the fact that convergence results forSLE’s involve the study ofmartingale observables, which are often discrete har- monic or discrete holomorphic functions. In the case of simply-connected domains, explicit expression for those are usually available, which is no longer the case in multiply connected domains.

Despite these difficulties, in [Zha08] the convergence of planar loop-erased ran- dom walks was extended to the case of multiply-connected domains. Multiple SLE’s were also defined and studied in various setups [BBK05, LK07, Dub07]. A natural requirement for the interface in the case of multiple marked points on the boundary is to be absolutely continuous with respect to the chordal SLE. It means that the driving force a_t describing each interface must be the Brownian motion √

κB_t pos- sibly with a drift (that may depend on the conformal moduli of the slit domain).

These drifts cannot be determined from Schramm’s principle only. In [BBK05], it was proposed to study the drifts that are related to certain CFT “correlations” (or partition functions). In [LK07, Dub07], some consistency conditions were imposed to determine possible values of the drifts. Recently, a definition of SLE for arbi- traryκ in multiply-connected domains was proposed in [Law11], on the grounds of restriction properties ofSLE.

In the present thesis, we consider a finitely-connected bounded domain Ω⊂ C with 2k+ 2 marked pointsa= a₀, a₁, . . . , a_2k, a_2k+1 = b on the boundary. Denote by∂_aΩ the boundary component containinga. We consider discrete approximations Ω^δ, a^δ, A^δ, b^δ, where A^δ := {a^δ₁, . . . , a^δ_2k} to the domain Ω, and a low-temperature expansion of the Ising model restricted to those configurations whose boundary is (a^δ, A^δ, b^δ). The configuration set in this expansion consists of subsetsSof edges and

boundary half-edges, such that any vertex is incident to an even number of edges in S, and that a^δ, A^δ, b^δ are the only boundary half-edges in S. These boundary conditions (insertion of boundary change operators ina^δ, A^δ, b^δ) are most natural in the low-temperature expansion. In the case of simply-connected domain, they cor- respond to “generalized Dobrushin” boundary conditions with multiple alternating arcs of pluses and minuses. For their meaning in the spin representation in general, see Subsection 2.1.

Any configuration in our model may be decomposed (not necessarily in a unique way) into a collection of loops andk+1 interfaces connecting the pointsa, a₁. . . , , a_2k, b in some order. We look at the interfaceγ^δ starting at the pointa^δ on the boundary component ∂aΩ^δ (note that we do not specify a priory at which point will it end up). In order to describe its scaling limit, we consider a conformal mapg₀^δ from Ω^δ to a nice sub-domain Λ^δ of the upper half-planeH, such thatg^δ₀(∂aΩ^δ) =R. We en- code the image K_t^δ:=g^δ₀(γ^δ) by chordal Loewner evolution in the upper half-plane, that is, parametrize K_t^δ by twice the half-plane capacity and consider the maps g^δ_t :=g_Kδ

t ◦g₀^δ, whereg_Kδ

t are conformal maps fromH\K_t^δ toHwith hydrodynamic normalization at infinity. These maps satisfy Loewner equation

∂tg^δ_t(z) = 2 g_t^δ(z)−a^δ_t,

where a^δ_t is a continuous driving function. We now describe our main result: the convergence of a^δ_t. Let, as δ → 0, the discrete domains Ω^δ, a^δ, A^δ, b^δ tend to a continuous domain Ω, a, A, bin a suitable sense (see (i^◦) – (iii^◦) and (iii^◦)Ain Section 3.1 for precise conditions). Let alsog₀^δ be chosen coherently for different δ, that is, converge to a conformal mapg₀ : Ω→Λ uniformly on compact subsets of Ω (and up to the boundary in the neighborhoods ofA,b). Letcbe a crosscut in Λ that separates g₀(a) from the images of other marked points and other boundary components, and let T^δ be the first time that γ^δ hits c. We denote by (Λ_t, a_t, A_t, b_t) the conformal image of (Ω, a, A, b) under the map gt.

Theorem 1.1. As δ → 0, a^δ|_[0,Tδ] converges in law to the solution a_[0,T] to the following SDE:

da_t=√

3dB_t+D_tdt, a₀ =g₀(a)

where the driftD_t:=D(Λ_t, a_t, A_t, b_t) = 3∂_alogZ(Λ_t, a_t, A_t, b_t). The functionZ(Λ, a, A, b) is the scaling limit of the partition function of the Ising model in domain Λ with boundary change operators at (a, A, b), and can be expressed in terms of the unique solution to certain boundary value problem (see (4.6), a^◦– e^◦and (3.20)). The law on the Loewner chains driven by at is conformally invariant.

Remark 1.2. (1) Although the notion of scaling limitZof the Ising partition function might be ambiguous, to make sense of ∂alogZ one only needs to know the scaling limits of ratios of partition functions with different positions of a, which is well- defined and conformally covariant.

(2) The convergence above should be understood in the sense that there is a coupling of a^δ_t and at, such that the stopping time T for at (when the solution crosses c) is close to the stopping timeT^δforv_t, and|a^δ_t−a_t|is uniformly small on their common domain of definition with probability tending to 1.

There are few cases when the scaling limits of the observables, and hence the drifts of driving process, can be computed explicitly. Thus, in the case of multiple

interfaces in simply-connected domains, the drift is an (explicit) rational function of the marked points at, At, bt ∈ R. In the case of four marked points, we recover the multiple SLE that previously appeared in the literature [BBK05] and prove rigorously the corresponding formula for crossing probabilities. In the case of an annulus, one can express the drifts in terms of Schwarz kernel in the annulus, which in its turn can be expressed via θ-functions, or infinite series of rational fractions.

As a limiting case, we obtain chordal and radial SLE3; in the latter case for the rigorous treatment of convergence we use reversibility of radial SLE recently proven in [Zha10].

The proof of convergence of discrete interfaces to SLE curves always relies on the existence of martingale observables, and their uniform convergence to confor- mally invariant or conformally covariant limits as the mesh size tends to zero. In the Ising case, the role of this observable is played bydiscrete holomorphic fermion F(Ω^δ, a^δ, z^δ) proposed by Smirnov [Smi06]. This observable is a solution to certain discrete boundary value problem, which allows one to prove its convergence to the solution to corresponding continuous problem [CS09]. In our (more general) situa- tion, we employ variants of this observable introduced and studied in the first part of the thesis, and rely upon the convergence results proven therein.

The same discrete holomorphic fermions can be used to express ∂_alogZ ap- pearing in the statement of the theorem. Indeed, if b^δ ∈ ∂aΩ, then the function

|F(Ω^δ, a^δ, b^δ)| is actually equal to the partition function of the Ising model with boundary change operators ata^δ andb^δ. Ifb^δ∈/ ∂_aΩ, then the corresponding parti- tion function can be expressed viaspinor version of the same observable, proposed in [ChI11]. Finally, in case of multiple boundary points one can express the partition function in terms of multi-point observable, which in its turn can be expressed in terms of the basic observables via a pfaffian formula. In order for the observable to express the partition function, we have to choose the cover$appropriately, namely, to branch around each boundary component that contains an odd number of marked points.

The computation of the scaling limit of the observable with martingale property is the main ingredient of our result. The rest of the proof closely follows one imple- mented by Zhan [Zha08] for the convergence loop-erased random walks in multiply- connected domains. It is fair to say that since that work (and the earlier work regarding simply-connected domains [LSW04]) it is known that the uniform conver- gence of holomorphic of harmonic martingale observable implies the convergence of driving forces of interfaces. In the most general situation of multiply-connected do- mains, the explicit expression for the observable is usually unavailable or untreatable;

however, in order to identify the scaling limits, it turns out to be sufficient to know the conformal covariance properties of the observable and its asymptotic behavior near the marked point a. The proof is rather technical and involves many details.

An important trick, also proposed in [Zha08], is the use of compactness arguments with respect to Carath´eodory-like topology in order to get uniform estimates from non-uniform ones.

1.3 Organization of the thesis

The thesis is organized as follows. Section 2 is devoted to the study of holomorphic spinor observables, their convergence to continuum limits and relation to partition functions and spin correlations. In Subsection 2.1, we give precise definitions of what will be meant by discrete domains and recall the definition of the critical Ising

model in low-temperature expansion. We also define specific boundary conditions we will deal with in the present thesis. In Subsection 2.2, we define basic holo- morphic spinor observables and prove that they are discrete holomorphic solution to certain (discrete) boundary value problem. We generalize the definition to the case of multiple marked points on the boundary in Subsection 2.3. We also prove a pfaffian formula that allows one to express these “multi-source” observables in terms of basic ones. In Subsection 2.4 we discuss relation of the observables to spin correlations and partition functions.

We state main convergence theorems for the basic observables in Subsection 3.1 and prove them in 3.2. In Section 3.3, we extend these results to the case of multi-source observables and discuss some properties of the limits. Further, we discuss applications of these results to convergence of certain ratios of spin correla- tions in Subsection 3.4, and provide the explicit formulae for spin correlations with Dobrushin boundary conditions in the half-plane in Subsection 3.5.

We discuss the martingale property of observables in Subsection 4.1. In Subsec- tion 4.2, we describe a way to derive heuristically the continuous driving processa_t, assuming that it is a continuous semi-martingale and that the scaling limit of the discrete martingale observable is a martingales with respect to the curve generated by a_t. We find that under these conditions, a_t is always equal to √

3B_t+Rt

D_sds, where Bt is a Brownian motion, and the drift Dt can be expressed in terms of the scaling limit of the observable. In Subsection 4.3, we give explicit expressions for the drift in simply-connected domain with multiple marked boundary points, and in doubly-connected domains. We also compute crossing probabilities for the limiting process in the case of four marked points, thus recovering the formula obtained in [BBK05].

In Section 5, we promote these non-rigorous computations to a rigorous proof, using Skorokhod embedding technique proposed in [LSW04]. This include a proof of numerous technical statements concerning the scaling limits of the observable.

Subsection 5.4 is a collection of concluding remarks. We start by observing absolute continuity of the limiting processes with respect to chordal SLE₃. We then briefly discuss convergence of the interfaces in a stronger topology and finish the rigorous proof of the “lattice model” counterpart of four-point crossing formula from Sub- section 4.3. We conclude by sketching the proof of convergence of the interface in doubly connected domain between outer boundary component and a shrinking inner boundary component to radial SLE3.

1.4 Acknowledgments

In the first place, I would like to thank my adviser, professor Stanislav Smirnov who introduced me to the field of two-dimensional lattice models and Schramm-Loewner evolutions, and shared much of his knowledge and ideas. I also appreciate his style of doing mathematics. He is a mathematician who always chooses difficult, interesting and important problems, and always keeps optimistic about solving them. I am also grateful to professor Victor Havin who directed my first steps in mathematical research, to many great lecturers from St. Petersburg State University, and to my high school teacher Natalya Efimova.

I enjoyed very much the collaboration with Dmitry Chelkak, who contributed to a large part of results of the thesis. It is a great pleasure to work together with such a strong and dedicated mathematician. I would like to thank all my colleagues in Geneva for creating a great mathematical environment. Special thanks to my

co-authors Kalle Kyt¨ol¨a, Dmitri Beliaev, Anton Alekseev and Andrey Bytsko, and my fellow graduate students Cl´ement Hongler and Hugo Duminil-Copin.

I am grateful to Denis Bernard, David Cimasoni and Ilya Gruzberg who kindly agreed to be the referees of the present thesis. David made a number of useful remarks concerning the preliminary version of the thesis.

During my graduate studies in Geneva, I was supported by EU RTN CODY, ERC AG CONFRA, and the Swiss National Science Foundation. Part of this work was done in the Chebyshev Laboratory, Faculty of Mathematics and Mechanics, St.

Petersburg State University, under the grant 11.G34.31.0026 of the Government of Russian Federation.

2 Spinor observables and their discrete holomorphicity

2.1 Notation and conventions

By(bounded) discrete domain(of meshδ) Ω^δwe mean a (bounded) connected subset of the square lattice Γ^δ =δZ² (example of a discrete domain is given on Fig. 1A).

More precisely, a discrete domain is specified by three sets V(Ω^δ) (vertices), F(Ω^δ) (faces) and E(Ω^δ) = Ein(Ω^δ)∪Ebd(Ω^δ) (interior edges and boundary half-edges, respectively), with the following requirements:

• all four edges and vertices incident to any facef∈ F(Ω^δ) belong toE(Ω^δ);

• every vertex in V(Ω^δ) is incident to four edges or half-edges in E(Ω^δ);

• every vertex that is incident to at least one edge or half-edgee∈E(Ω^δ) belongs toV(Ω^δ);

We will sometimes also impose the following condition:

at least one of two faces incident to any edge

e∈Ein(Ω^δ) belongs to F(Ω^δ). (2.1) This requirement is convenient in the proof of convergence results, so we will assume it throughout the corresponding chapters. However, when dealing with the interfaces, we will not be able to afford this condition, since domains obtained by cutting out the interface need not satisfy it. It turns out, however, that the convergence for such “nice” domains is sufficient for convergence of the interfaces (see Lemma 4.1).

For an interior edge e∈E_in(Ω^δ) we denote byz_e its midpoint. For a boundary half-edge e ∈ Ebd(Ω^δ) we denote by ze its endpoint which is not a vertex of Γ^δ. When no confusion arises we will identify an edge (or half-edge) ewith a pointze.

By the boundary ∂Ω^δ of Ω^δ we will mean the set of all its boundary half-edges E_bd(Ω^δ) or, if no confusion arises, the set of corresponding endpointsz_e.

We will need a notion of double covers of the graph Ω^δ. Adouble cover ofΩ^δ is a graphΩe^δ with a two-to-one local graph isomorphism$:Ωe^δ→Ω^δ. Given a marked boundary half-edge a∈∂Ω^δ, we will describe points z on a double cover by lattice paths γ running from a toz in Ω^δ, modulo homotopy and modulo an appropriate subgroup of the fundamental group. If Ω^δ is k-connected, there are 2^k−1 subgroups that give rise to double covers, leading to 2^k−1 covers (including trivial one). In other words, for each hole one should specify whether or not walking around this hole changes the sheet. Ifzis a point on a double coverΩe^δ, we letz^∗ ∈Ωe^δbe defined by $(z^∗) =$(z), z^∗ 6=z. We will also use the obvious notation V(Ωe^δ), E(Ωe^δ) etc.

We will work with the low-temperature contour representation of the critical Ising model in Ω^δ (see [Pal07]). We call a subset S of edges and half-edges in Ω^δ (see Fig. 1B, note that we admit inner half-edges in S) a generalized configuration or a generalized interfaces picture for this model, if

• each vertex in Ω^δ is incident to 0,2 or 4 edges and half-edges inS;

• if an edge e= e⁰ ∪e⁰⁰ consists of two halves e⁰, e⁰⁰, then at most one of those three belongs to S.

We will denote the set of all generalized configurations in Ω^δ by Conf(Ω^δ). By the boundary ∂S of S ∈ Conf(Ω^δ) we will mean the set of all half-edges e ∈S (or the corresponding pointsz_e, if no confusion arises). Thepartition functionof the critical Ising model is given by

Z_free(Ω^δ) = X

S∈Conf_free(Ω^δ)

x^|S|, x=xcrit =√

2−1. (2.2)

Here and below|S|is the total number of edges and half-edges in S, and Conf_free(Ω^δ) :={S∈Conf(Ω^δ) :∂S⊂∂Ω^δ}.

The value x =xcrit will be fixed throughout the paper. The formula (2.2) endows the set Conf_free(Ω^δ) of configurations with a probability measure, the probability of a particular configurationS beingx^|S|/Z_free(Ω^δ).

The subscript “free” in the notation means that this configuration set corre- sponds to free boundary conditions in the spin representation. We will, however, mostly work with subsets of Conf_free(Ω^δ) and restrictions of the probability measure to those subsets. Thus, we denote

Conf+(Ω^δ) :={S ∈Conf(Ω^δ) :∂S=∅}

Confe1,...,en(Ω^δ) :={S∈Conf(Ω^δ) :∂S={z_e1, . . . zen}mod 2}, (2.3) meaning that if some points (edges or half-edges)e₁, . . . , e_n∈E(Ω^δ) appear several times in the subscript, we keep only those appearing an odd number of times.

The low-temperature expansion is naturally mapped onto the spin representa- tion of the Ising model, where the interfaces are boundaries between “+” and “−”

spins on faces (fixing the spin at some face f yields one-to-one correspondence).

In the case of simply-connected domains, the configuration set Confe1,...,e2k(Ω^δ) de- fined as above corresponds to configurations of spins with alternating “+” and “−”

boundary conditions on boundary arcs (e_j, e_j+1). In the multiply-connected case the situation is more complicated. If there is an even number of marked points ej

on each boundary component, then restricting to Conf_e1,...,e2k(Ω^δ) corresponds to fixing the spins along one boundary component only, while for all other components it imposeslocally monochromatic conditions, i. e. requires the spins to be constant (either “+” or “−”) along each boundary arc (e_j, e_j+1), and to change at marked points. If there are components carrying an odd number ofe_j, then Conf_e1,...,e2k(Ω^δ) cannot correspond to the spin representation of the Ising model on the graph Ω^δ, since going around such a component we intersect the interface an odd number of times, and thus arrive at the opposite spin. One can, however, define the configu- ration set of the model to be an assignment of spins to the faces thedouble cover Ω˜^δ of the original graph, with the condition that two fibers of the same face in Ω^δ carry opposite spins. If we choose the double cover to branch exactly around those components that have an odd number ofej, this representation becomes compatible with boundary conditions.

2.2 Basic observables

In this section we will construct spinor observables and prove their discrete holo- morphicity. These observables should be considered as natural generalizations of

Figure 1: (A) An example of four-connected discrete domain Ω^δ. (B) An example of generalized configuration (or generalized interfaces picture) S ∈ Confa,z(Ω^δ), decomposed into a collection of loops and a simple lattice path γ : a z, as required in the Definition 2.1

fermionic observables introduced by Smirnov [Smi06], [CS09] to the multiply con- nected setup. The discrete domain Ω^δ and a boundary half-edge a ∈ ∂Ω^δ will be fixed throughout this section. In order to give a consistent definition for all discrete domains, we need the following (technical) notation. A boundary half-edge a, ori- ented from za to the inner vertex, can be thought of as a complex number. Then we set

η_a :=e^{−i2(arg(a)+12π)}= (ia/|a|)⁻¹² (2.4) for some fixed choice of the sign. Note that, if ais south-directed, then ηa=±1.

Given a point z_e (i.e., an edge or a boundary half-edge e ∈ E(Ω^δ)), a configu- ration S ∈Confa,e(Ω^δ), a double cover$ :Ωe^δ → Ω^δ, and a point z ∈ $⁻¹(ze) on this double cover, we introduce thecomplex phase of S with respect toz. First, we decompose S into a collection of loops and a path γ running from a to e so that there are no transversal intersections or self-intersections (see Fig. 1B). Then, we denote by l(S) the number of non-trivial loops in S (that is, those loops for which one changes the sheet ofΩe^δ when going around the loop). We also introduce a sign s(z, S) := +1, if the pathγ describesz, ands(z, S) :=−1, ifγ describesz^∗.

Definition 2.1. The complex phase of a configuration S ∈ Conf_a,$(z)(Ω^δ) with respect to a point z lying on a double cover $:Ωe^δ →Ω^δ is defined as

W$(z, S) :=e^−i2w(γ)·(−1)^l(S)·s(z, S), (2.5) wherew(γ)denotes the winding (i.e., the increment of the argument) alongγ. Then, we define a spinor observable on the double cover Ωe^δ as

F$(Ω^δ, a, z) := X

S∈Conf_a,$(z)(Ω^δ)

ηaW$(z, S)x^|S|. (2.6) Remark 2.2. (i)W_$(z, S) does not depend on the way one chooses the decomposition of a given configuration S. The proof is elementary, and we leave it to the reader.

Note that it is sufficient to check that the second factor (−1)^l(S)s(z, S) is well-defined, as the rest is well-known (e.g., see Lemma 7 in [HS10]).

(ii) By definition,F$(Ω^δ, a, z^∗)≡ −F_$(Ω^δ, a, z), thus we callF$(a, z) aspinor.

(iii) If $ is the trivial cover, then Definition 2.1 reproduces the original construc- tion due to Smirnov (e.g., see (2.10) in [CS09]). We denote this (non-branching) observable byF₀(Ω^δ, a, z) and the corresponding complex phase byW₀(z, S).

(iv) We will usually omit Ω^δ in the notation unless needed.

In order to claim that our observables arediscrete holomorphic, we introduce the following notation. Given a vertex v ∈ V(Ω^δ), we consider four nearby corners of faces incident tov, and identify them with the points v_k:=v+e^iπ(2k+1)/4·δ/2√

2, k= 0,1,2,3. We denote sets of all corners of Ω^δ and its double cover Ωe^δ by Υ(Ω^δ) and Υ(eΩ^δ), respectively. Similarly to (2.4), for a corner c=v_k∈Υ(Ωe^δ) we set

ηc:= (i(c−v)/|c−v|)⁻¹² :=e−iπ(2k+1)/8

(again, the particular choice of square root signs is unimportant, so we fix it once forever for each of four possible orientations ofc−v). We denote by

Pr_ηc(F) := Re(η_cF)η_c= ¹₂(F +η_c²F)

the orthogonal projection of a complex numberF ∈Conto the line ηcR.

The main theorem concerning the “discrete” properties of the observable (2.6) states its s-holomorphicity (see Section 3 in [Smi10] or Definition 3.1 in [CS09]) and describes the boundary conditions:

Theorem 2.3. For any corner c ∈ Υ(eΩ^δ) formed by edges or half-edges z⁰, z⁰⁰ ∈ E(Ω^δ), one has

Prηc(F$(a, z⁰)) =Prηc(F$(a, z⁰⁰)). (2.7) Moreover, ifb∈∂Ωe^δ\ {a, a^∗} is a boundary half-edge, thenF$(a, b)kiη_b, i.e.,

Prη_b(F$(a, b)) = 0, (2.8)

where η_b is defined (up to a sign which is unimportant for us) similarly to (2.4).

Remark 2.4. Since our observables are spinors (i.e., F_$(a, z^∗) ≡ −F_$(a, z)), the identities (2.7) at two corners c, c^∗ such that $(c) = $(c^∗) are equivalent. The same is fulfilled for the boundary condition (2.8).

Proof. Let v denotes the vertex incident to both z⁰ and z⁰⁰. There exists a natural bijection Π : Conf_a,$(z⁰₎(Ω^δ) → Conf_a,$(z⁰⁰₎(Ω^δ), provided by taking “xor” of a generalized configuration S with two half-edges (z⁰v) and (z⁰⁰v). The well-known proof of the theorem for the trivial cover (see, e.g. Proposition 2.5 in [CS09] or Lemma 45 in [HS10]) assures that, for anyS ∈Conf_a,$(z⁰₎(Ω^δ),

Pr_ηc(W₀(z⁰, S)x^|S|) =Pr_ηc(W₀(z⁰⁰,Π(S))x^|Π(S)|).

Clearly, the same holds true withW₀replaced byW_$, unless Π changes the number of non-trivial loopsl(S) ors(S, z⁰)6=s(Π(S), z⁰⁰). However, it is easy to see that Π always preserves the factor (−1)^l·s: for instance, if there was a non-trivial loop in Sthat disappeared in Π(S), then this loop has become a part of the pathγ, leading to the simultaneous change of the signs.

To derive the boundary condition (2.8), it is sufficient to note that the winding of any curveγ running fromatob is equal to (argb+π)−argamodulo 2π.

Remark 2.5. A careful examination of the proof shows that it uses in a crucial way that the winding of any simple planar loop is an odd multiple of 2π. This is why we forbid transversal self-intersections in the decomposition.

2.3 Multi-source observables and pfaffian formulae

In this section we modify the definition of observables to the case of multiple marked points on the boundary, and prove the Pfaffian formula (similar to the one obtained in [Hon10] in the case of non-branching observables) that allow one to express this multi-point observables in terms of the basic ones. Let boundary half-edges a = a0, a1, . . . , a_2k belong to the boundary of a double cover Ωe^δ; denote

A:={a₁, . . . , a_2k},

and letz∈Ωe^δ be an interior edge or a boundary half-edge. To each half-edgeai we assign a complex number η_ai according to (2.4) with some choice of the signs which will be fixed from now on. Define

η^a_aj^s :=iηaj

η_assign(j−s).

We note here that, up to a sign, this is equal toe^−iw/2, where w is the winding of any curve in Ω^δconnecting a_l toa_j. Connect the pairs of half-edges (a₁;a₂),(a₃;a₄), . . . by artificial arcsl12, l34, . . ., as shown on Fig 2.3. We also note thate^{−iw(l2j−1,2j)/2} =

±η_a^a2j−1_2j , and we assume the sign would be “−” (this is a matter of choice ofnaj of arcs l2j−1,2j).

Given a generalized configuration S ∈ Conf_{a0,a1,...,a2n,z}(Ω^δ), or, in other words, a generalized interfaces picture S such that ∂S = {a₀, a1, . . . , a_2k, z}, we can de- compose it into a collection of loops and simple curves connecting a_j and zat some order. Adding artificial arcs l2j−1,2j, we obtain a single curve from a to z, and, possibly, some more loops (see Fig. 2.3). For such configuration, we define l(S) to be the total number of loops that change a sheet of$ plus a total number of loops whose winding is an integer factor of 4π (such loops might exist since the artificial arcs may create self-intersections). Next, use the formula (2.5) in order to define W_$(S) and generalize definition (2.6):

F_$(a₀, A;z) =F_$(a₀, a₁, . . . , a_2k;z)

: = X

S∈Conf_$(a

0),$(a1)...,$(a2k),$(z)(Ω^δ)

η_a0W_$(S)x^|S|. (2.9) Proposition 2.6. The observables (2.9) are s-holomorphic spinors satisfying the boundary condition (2.8) everywhere on ∂Ω^δ except the points a₀, a₁, . . . , a_2k. Proof. We use the same argument as in the proof of Theorem 2.3. The only difference is that, when applying the bijection Π, we may destroy a (self-intersecting) loop whose winding is an integer multiple of 4π, and when that loops becomes part of γ, its contribution to e^−iw(γ)/2 has opposite sign as compared to the simple loop.

However, our modified definition of l(s) exactly accounts for that effect.

In accordance with our convention “mod 2” in the definition (2.3), this formula also defines the values ofF_$ at marked boundary points, i.e. forz=a₀, a₁, . . . , a_2k. In that case∂Sconsists of 2kpoints instead of 2k+2, so one can expressF$(a, A, aj) in terms of the observable with smaller number of marked points. Careful examina- tion of the windings provides the following formulae (see also Fig. 2.3):

F$(a0, A;a0) =ηa0(ηa1η_a^a1₂)⁻¹F$(a1, A\ {a₁, a2};a2) ; F_$(a₀, A;a_2s) =−η^a_a^2s−1

2s F_$(a₀, A\ {a_2s−1, a_2s};a2s−1) ; F_$(a₀, A;a2s−1) =−η^a_a^2s

2s−1F_$(a₀, A\ {a_2s−1, a_2s};a_2s).

(2.10)

Figure 2: (A) A doubly connected domain Ω^δ and a generalized configuration S ∈ Conf_{a,a1,...,a6,z}(Ω^δ). Adding artificial arcs (a₁;a₂),(a₃;a₄),(a₅;a₆) (dashed lines), we promoteS to a collection of loops and a simple pathγ running form a₀ toz. Note that the loop containing a5, a6 is self-intersecting and its winding is zero, which contributes a (−1) factor into the weight of the configuration. (B) A configuration contributes toF_$(a, A, a₃) if and only if it contributes to F_$(a, A\{a₃, a₄}, a₄), and the contributions differ by a factor ofe^−iw(l43)/2.

The following recurrent relation together with the substitution rules (2.10) allows us to express all observables (2.9) in terms of the original observable (2.6).

Lemma 2.7. The identity

F_$(a₀, A;z) =

2k

X

j=0

F$(a0, A;aj)

F_$(a_j;a_j) F_$(a_j;z) (2.11) is fulfilled for any z∈Ωe^δ.

Proof. Both sides of (2.11) are discrete s-holomorphic spinors (defined on the same double cover) satisfying the boundary condition (2.8) everywhere on∂Ωe^δexcept the marked pointsa0, . . . , a2n. Moreover, for anyj= 0, . . . ,2k, there is only one term in the sum (2.11) that fails to satisfy (2.8) ata_j. However, its value ata_j coincides with the left-hand side valueF_$(a₀, A;a_j). Hence, these two s-holomorphic spinors are equal to each other due to Remark 3.6, since their difference satisfies the boundary condition (2.8)everywhere on ∂Ωe^δ.

We introduce some notation. Let G^δ_i,j := F$(Ω^δ, ai;aj)

η_a^ai_jF_$(Ω^δ, a_i;a_i) 0≤i6=j≤2k G^δ_i,2k+1=G^δ_2k+1,i:=F_$(Ω^δ, a_i;z) 0≤i≤2k

G^δ_i,i = 0, 0≤i6=j≤2k+ 1.

Remark 2.8. (i) Note that F_$(Ω^δ, a_i;a_i)/η_ai does not depend on i, and it follows from positivity of spin correlations that it is never equal to zero.

(ii) It follows from the definition thatG^δ_l,j =G^δ_j,l. Indeed, due to the first remark, it is sufficient to show that ^F_iη^{$(Ωδ,al;aj)}

ajsgn(j−l) = ^F_iη^{$(Ωδ,aj;al)}

alsgn(l−j), and this can be seen by examining a contribution of eachS∈Confai,aj(Ω^δ) to both sides.