We prove convergence of multiple interfaces in the critical planar q = 2 random cluster model, and provide an explicit description of the scaling limit. Remarkably, the expression for the partition function of the resulting multiple SLE(16/3) coincides with the bulk spin correlation in the critical Ising model in the half-plane, after formally replacing a position of each spin and its complex conjugate with a pair of points on the real line. As a corollary, we recover Belavin-Polyakov-Zamolodchikov equations for the spin correlations.
In this paper, we show that the interfaces in FK Ising model in any domain with 4 marked boundary points and wired--free--wired--free boundary conditions conditioned on a specific internal arc configuration of interfaces converge in the scaling limit to hypergeometric SLE (hSLE). The arc configuration consists of a pair of interfaces and the scaling limit of their joint law can be described by an algorithm to sample the pair from a hSLE curve and a chordal SLE (in a random domain defined by the hSLE).
We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter $κ=3$ and $κ=16/3$ respectively.
In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the …
We present a mathematical proof of theoretical predictions made by Arguin and Saint-Aubin, as well as by Bauer, Bernard, and Kytola, about certain non-local observables for the two-dimensional Ising model at criticality by combining Smirnov's recent proof of the fact that the scaling limit of critical Ising interfaces can be described by chordal SLE(3) with Kozdron and Lawler's configurational measure on mutually avoiding chordal SLE paths. As an extension of this result, we also compute the probability that an SLE(k) path, k in (0,4], and a Brownian motion excursion do not intersect.
Schramm Loewner Evolution (SLE) is a one-parameter family of random planar curves introduced by Oded Schramm in 1999 as the candidates for the scaling limits of the interfaces in the planar critical lattice models. This is the only possible process with conformal invariance and a certain "domain Markov property". In 2010, Chelkak and Smirnov proved the conformal invariance of the scaling limits of the critial planar FK-Ising model which gave the convergence of the interface to SLE$_{16/3}$. We derive the arm exponents of SLE$_κ$ for $κ\in (4,8)$. Combining with the convergence of the interface…
In the second article of this series, we establish the convergence of the loop ensemble of interfaces in the random cluster Ising model to a conformal loop ensemble (CLE) --- thus completely describing the scaling limit of the model in terms of the random geometry of interfaces. The central tool of the present article is the convergence of an exploration tree of the discrete loop ensemble to a branching SLE$(16/3,-2/3)$. Such branching version of the Schramm's SLE not only enjoys the locality property, but also arises logically from the Ising model observables.
We study the interfaces arising in the two-dimensional Ising model at critical temperature, without magnetic field. We show that in the presence of free boundary conditions between plus and minus spins, the scaling limit of these interfaces can be described by a variant of SLE, called dipolar SLE(3). This generalizes a celebrated result of Chelkak and Smirnov and proves a conjecture of Bauer, Bernard and Houdayer. We mention two possible applications of our result.
We review the algebraic and analytic aspects of the conformal field theory (CFT) and its relation to the stochastic Loewner evolution (SLE) in an example of the Ising model. We obtain the scaling limit of the correlation functions of Ising free fermions on an arbitrary simply connected two-dimensional domain $D$. Then, we study the analytic and algebraic aspects of the fermionic CFT on $D$, using the Fock space formalism of fields, and the Clifford vertex operator algebra (VOA). These constructions lead to the conformal field theory of the Fock space fields and the fermionic Fock space of stat…
We prove a general result on convergence of interfaces in the critical planar Ising model to conformally invariant curves absolutely continuous with respect to SLE(3). Our setup includes multiple interfaces on arbitrary finitely connected domains, and we also treat the radial SLE case. In the case of simply and doubly connected domains, the limiting processes are described explicitly in terms of rational and elliptic functions, respectively.
We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing the convergence of certain exploration processes towards SLE$(3,\frac{-3}2,\frac{-3}2)$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield.
In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally-invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishin…
We construct martingale observables for systems of multiple SLE curves by applying screening techniques within the CFT framework recently developed by Kang and Makarov, extended to admit multiple SLEs. We illustrate this approach by rigorously establishing explicit expressions for the Green's function and Schramm's formula in the case of two curves growing towards infinity. In the special case when the curves are `fused' and start at the same point, some of the formulas we prove were predicted in the physics literature.
This article pertains to the classification of multiple Schramm-Loewner evolutions (SLE). We construct the pure partition functions of multiple SLE$(κ)$ with $κ\in (0,4]$ and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from [BBK05, KP16]. We prove that the two approaches to construct multiple SLEs - the global, configurational construction of [KL07, Law09a] and the local, growth process construction of [BBK05, Dub07, Gra07, KP16] - agree. The pure partition functions are closely related to crossing probabilities in critical statistical mechanics models. W…
We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.
The scaling limit of the spin cluster boundaries of the Ising model with domain wall boundary conditions is SLE with kappa=3. We hypothesise that the three-state Potts model with appropriate boundary conditions has spin cluster boundaries which are also SLE in the scaling limit, but with kappa=10/3. To test this, we generate samples using the Wolff algorithm and test them against predictions of SLE: we examine the statistics of the Loewner driving function, estimate the fractal dimension and test against Schramm's formula. The results are in support of our hypothesis.
We consider collections of $N$ chordal random curves obtained from a critical lattice model on a planar graph, in the limit when a fine-mesh graph approximates a simply-connected domain. We define and study candidates for such limits in terms of conformally invariant collections of random curves, generated via iterated Loewner equations. These curves are a natural ``domain Markov extension'' of the earlier introduced local multiple SLE initial segments to global multiple SLE curves. For realizing them as scaling limits, we provide two a priori results to guarantee the precompactness of the dis…
These lecture notes provide a (almost) self-contained account on conformal invariance of the planar critical Ising and FK-Ising models. They present the theory of discrete holomorphic functions and its applications to planar statistical physics (more precisely to the convergence of fermionic observables). Convergence to SLE is discussed briefly. Many open questions are included.
We present basic properties of Dipolar SLEs, a new version of stochastic Loewner evolutions (SLE) in which the critical interfaces end randomly on an interval of the boundary of a planar domain. We present a general argument explaining why correlation functions of models of statistical mechanics are expected to be martingales and we give a relation between dipolar SLEs and CFTs. We compute SLE excursion and/or visiting probabilities, including the probability for a point to be on the left/right of the SLE trace or that to be inside the SLE hull. These functions, which turn out to be harmonic, …
We prove convergence results for variants of Smirnov's fermionic observable in the critical Ising model in presence of free boundary conditions. One application of our analysis is a simple proof of a theorem by Hongler and Kytölä on convergence of critical Ising interfaces with plus-minus-free boundary conditions to dipolar SLE(3), and generalization of this result to arbitrary number of arcs carrying plus, minus or free boundary conditions. Another application is a computation of scaling limits of crossing probabilities in FK-Ising model with arbitrary number of alternating wired/free boundar…
In this paper we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of $n$ slits that connect $n$ points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.
We consider the dimer model on piecewise Temperleyan, simply connected domains, on families of graphs which include the square lattice as well as superposition graphs. We focus on the spanning tree $\mathcal{T}_δ$ associated to this model via Temperley's bijection, which turns out to be a Uniform Spanning Tree with singular alternating boundary conditions. Generalising the work of the second author with Peltola and Wu \cite{LiuPeltolaWuUST} we obtain a scaling limit result for $\mathcal{T}_δ$. For instance, in the simplest nontrivial case, the limit of $\mathcal{T}_δ$ is described by a pair of…
In this paper, we consider the set of interfaces between + and - spins arising for the critical planar Ising model on a domain with + boundary conditions, and show that it converges towards CLE(3). Our proof relies on the study of the coupling between the Ising model and its random cluster (FK) representation, and of the interactions between FK and Ising interfaces. The main idea is to construct an exploration process starting from the boundary of the domain, to discover the Ising loops and to establish its convergence to a conformally invariant limit. The challenge is that Ising loops do not …
The present paper is concerned with properties of multiple Schramm--Loewner evolutions (SLEs) labelled by a parameter $κ\in (0,8]$. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson's Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the…
In this paper we give a physical interpretation of the probability of a Stochastic Loewner Evolution (SLE) trace approaching a marked point in the upper half plane, e.g. on another trace. Our approach is based on the concept of fusion of boundary with bulk or other boundary fields in Conformal Field Theory (CFT). In this context we also state a proposal how the results can be extended to situations that happen not sufficiently far away from the boundary.
Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and…
Multiple Schramm-Loewner Evolutions (SLE) are conformally invariant random processes of several curves, whose construction by growth processes relies on partition functions: Möbius covariant solutions to a system of second order partial differential equations. In this article, we use a quantum group technique to construct a distinguished basis of solutions, which conjecturally correspond to the extremal points of the convex set of probability measures of multiple SLEs.
Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensiona…
The Ising model at inverse temperature $β$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-β}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $β$, this generates a dependent (spin) percolation model with parameter $r$. We show…
We provide multiple Schramm-Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess-Zumino-Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE trac…
In the series of lectures, we will discuss probability laws of random points, curves, and surfaces. Starting from a brief review of the notion of martingales, one-dimensional Brownian motion (BM), and the $D$-dimensional Bessel processes, BES$_{D}$, $D \geq 1$, first we study Dyson's Brownian motion model with parameter $β>0$, DYS$_β$, which is regarded as multivariate extensions of BES$_D$ with the relation $β=D-1$. Next, using the reproducing kernels of Hilbert function spaces, the Gaussian analytic functions (GAFs) are defined on a unit disk and an annulus. As zeros of the GAFs, determinant…
We formulate multiple Schramm-Loewner evolutions (SLEs) for coset Wess-Zumino-Witten (WZW) models. The resultant SLEs may describe the critical behavior of multiple interfaces for the 2D statistical mechanics models whose critical properties are classified by coset WZW models. The SLEs are essentially characterized by multiple Brownian motions on a Lie group manifold as well as those on the real axis. The drift terms of the Brownian motions, which come from interactions of interfaces, are explicitly determined by imposing a martingale condition on correlation functions among boundary condition…
It is known that a backward Schramm--Loewner evolution (SLE) is coupled with a free boundary Gaussian free field (GFF) with boundary perturbation to give conformal welding of quantum surfaces. Motivated by a generalization of conformal welding for quantum surfaces with multiple marked boundary points, we propose a notion of multiple backward SLE. To this aim, we investigate the commutation relation between two backward Loewner chains, and consequently, we find that the driving process of each backward Loewner chain has to have a drift term given by logarithmic derivative of a partition functio…
The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on $\Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical g…
Recently del Monaco and Schleißinger addressed an interesting problem whether one can take the limit of multiple Schramm--Loewner evolution (SLE) as the number of slits $N$ goes to infinity. When the $N$ slits grow from points on the real line ${\mathbb{R}}$ in a simultaneous way and go to infinity within the upper half plane ${\mathbb{H}}$, an ordinary differential equation describing time evolution of the conformal map $g_t(z)$ was derived in the $N \to \infty$ limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation go…
We describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation and have the geodesic multichord property. These results are entirely self-contained and have…
The concept of Schramm-Loewner evolution provides a unified description of domain boundaries of many lattice spin systems in two dimensions, possibly even including systems with quenched disorder. Here, we study domain walls in the random-field Ising model. Although, in two dimensions, this system does not show an ordering transition to a ferromagnetic state, in the presence of a uniform external field spin domains percolate beyond a critical field strength. Using exact ground state calculations for very large systems, we examine ground state domain walls near this percolation transition findi…
In a previous article, we define "connectivity weights" to be functions with these two properties: 1) They solve the three conformal Ward identities of conformal field theory (CFT) and a system of $2N$ null-state differential equations governing a CFT $2N$-point function of $ϕ_{1,2}$ or $ϕ_{2,1}$ primary Kac operators. 2) They satisfy a certain "duality" condition. In that same article, we argue that these functions are in fact pure partition functions for a multiple-SLE$_κ$ process with $2N$ curves, and we show how to find explicit formulas for them in terms of Coulomb gas contour integrals. …
