Stochastic Loewner Evolution

January 24, 2006, IHP, Amphi Hermite

Moderator: Wendelin Werner (Orsay)

Michel Bauer (Saclay): Schramm-Loewner evolutions and conformal field theories

Since the mid-eighties, physicists have developed powerful field theory tools to cope with local aspects of scale invariant systems in two dimensions. However, a good understanding of non-local objects like interfaces in these systems has emerged only quite recently, based on stochastic processes coupled to conformal geometry via Loewner equations. In this talk,I shall review briefly the interplay between conformal field theories and Schramm-Loewner evolutions. At the heart of the correspondence is the relation between partition functions in statistical mechanics and martingales in probability theory.
If time permits, I shall give some applications and generalizations.

Vincent Beffara (ENS, Lyon): Critical percolation on regular triangulations

We investigate possible ways to generalize Smirnov's proof of Cardy's formula for crossing probabilities in the triangular lattice, to other bi-periodic triangulations of the complex plane.

Federico Camia (Amsterdam): The scaling limit of two-dimensional critical percolation

In this talk I will consider critical site percolation on the triangular lattice, which is simply the random black or white coloring of a regular hexagonal tiling of the plane. Scaling the diameter of the hexagons to zero and focusing on the boundaries between black and white clusters reveals a complex geometric structure, with the appearance of conformally invariant, random, fractal curves. In recent years, substantial progress has been made in understanding the properties of these curves in terms of the Stochastic Loewner Evolution introduced by Schramm. I will describe some of this progress, including recent work done in collaboration with C. M. Newman and with Newman and L. R. Fontes, which gives a complete picture of the scaling limit of critical percolation and a possible framework to study the scaling limit of near-critical percolation and of related models.

John Cardy (Oxford): Conformal restriction and the stress tensor of conformal field theory

For SLE(8/3) and other processes satisfying conformal restriction, we identify the stress tensor of conformal field theory and show that it satisfies the expected Ward identities. (This work was done in collaboration with B Doyon and V Riva.)

Jesper Lykke Jacobsen (Orsay): Random curves and antiferromagnetism

Seen from a physicist's point of view, SLE${}_\kappa$ basically describes the critical fluctuations of domain walls in the O($n$) model, or of cluster boundaries in the $Q$-state Potts model, directly in the continuum limit. Within Conformal Field Theory (CFT), both of the latter models have traditionally been described in the continuum limit through a one-component Coulomb gas.
In recent years, much effort has been devoted to generalise SLE to describe other systems enjoying conformal invariance, and to make clearer the connection with CFT. In principle, CFT and the theory of integrable systems offer many conformally invariant systems whose description goes beyond the one-component Coulomb gas, and thus, could serve as inspiration for the SLE community. But in most cases, either there is no obvious connection to random curves, or the random curves are scaling limits of a microscopic model which is non-universal in the sense that it makes essential use of the underlying lattice structure.
Here we present recent results on a lattice model of random curves whose continuum limit is independent of the underlying lattice and necessitates a two-component Coulomb gas. It describes the cluster boundaries of the $Q$-state Potts model at the {\em antiferromagnetic} phase transition. This transition presents several unusual features. First, it is a so-called first-order critical point, i.e., it is simultaneously first and second order. Second, one of the Coulomb gas components is non-compact, i.e., it contributes a continuous part to the spectrum of critical exponents. Third, whenever $Q$ is a Beraha number, i.e., $Q=4 \cos^2 \frac{\pi}{n}$ with $n$ integer, the physics is profoundly different from the case of generic $Q$.
(Work done in collaboration with H. Saleur.)

Stanislav Smirnov (Genève & Stockholm): TBA

Jose A. Trujillo Ferreras (Zürich): The expected area of the filled Brownian loop is Pi/5

Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion conditioned so that B_0 = B_1) and consider the compact hull obtained after filling in all the holes, i.e., the complement of the unique unbounded component of R^2\B[0,1]. The expected area of this hull is Pi/5. Even though the statement of the problem is very classical, the computation of the expected area uses much more recent techniques: conformal restriction measures and SLE (Schramm Loewner Evolution). In this talk, I will explain how one can use these new tools to get at the expected area of the Brownian loop; I will give a brief overview of the necessary background.
If time allows I will also explain how one can use a result of Yor about the law of the index of a Brownian loop to show that the expected areas of the regions of non-zero index n equal 1/(2 Pi n^2). As a consequence, one finds that the expected area of the region of index zero inside the loop is Pi/30; a quantity that could not be computed directly using Yor's index description.