Related papers: Random curves, scaling limits and Loewner evolutio…
This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Loewner Evolution (SLE) by Oded…
We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition…
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models.…
This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions…
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…
The Stochastic Loewner equation, introduced by Schramm, gives us a powerful way to study and classify critical random curves and interfaces in two-dimensional statistical mechanics. New kind of stochastic Loewner equation, called fractional…
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…
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…
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…
Stochastic Loewner evolution also called Schramm Loewner evolution (abbreviated, SLE) is a rigorous tool in mathematics and statistical physics for generating and studying scale invariant or fractal random curves in two dimensions. The…
Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability distributions are parameterized by a…
This article focuses on the characterization of global multiple Schramm-Loewner evolutions (SLE). The chordal SLE describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and…
We point out that the construction of a martingale observable describing the spin interface of the two-dimensional Ising model extends to a class of non-integrable variants of the two-dimensional Ising model, and express it in terms of…
Loop-erased random walk and it's scaling limit, Schramm--Loewner evolution, have found numerous applications in mathematics and physics. We present a 2 dimensional analogue of LERW, the loop erased random surface. We do this by defining a 2…
These lecture notes on 2D growth processes are divided in two parts. The first part is a non-technical introduction to stochastic Loewner evolutions (SLEs). Their relationship with 2D critical interfaces is illustrated using numerical…
The scaling limit of planar loop-erased random walks is described by a stochastic Loewner evolution with parameter kappa=2. In this note SLE(2) in the upper half-plane H minus a simply-connected compact subset K of H is studied. As a main…
Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling…
The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the parameter kappa. These lattice models have a natural parametrization of their random…
We provide a general framework of estimates for convergence rates of random discrete model curves approaching Schramm Loewner Evolution (SLE) curves in the lattice size scaling limit. We show that a power-law convergence rate of an…
The development of Schramm--Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde \cite{MR05}, is…