Conformally invariant scaling limits: an overview and a collection of problems

Abstract

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. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.

Boundary partitions in trees and dimers

“…Carleson noticed that Cardy's formula is especially nice when the domain is an equilateral triangle, and this was one of the insights that led to Smirnov's proof that the percolation interface converges to SLE 6 [Smi01] (see also [CN07]). (The SLE κ process, which we do not define here, was introduced by Schramm [Sch00] and describes the scaling limits of random curves arising in statistical physics; see [Sch07].) Arguin and Saint-Aubin [ASA02, § 3] gave (a physics derivation of) the corresponding crossing probabilities for spins of the critical Ising model, shown in In a related vein, there has been much study of the physics of multiple-strand networks of self-avoiding walks (polymers), loop-erased random walks, and other random paths that are either known or believed to be related to SLE κ .…”

The Fourier spectrum of critical percolation

“…Here, the o(1) represents a function of λ and R that tends to 0 as λ → 0, uniformly in R. This answers Problem 5.1 from [Sch07]. Even for the square lattice, (1.3) follows from Theorem 1.1 combined with (1.5).…”

“…This has to do with the scaling limit of dynamical percolation, as introduced in [Sch07], and whose existence we plan to show in [GPS]. In this scaling limit, time and space are both scaled, and the relationship between their scaling is chosen in such a way that the event of the existence of a percolation crossing of the unit square at time 0 and at one unit of time later have some fixed correlation strictly between 0 and 1.…”

John Michael Hammersley. 21 March 1920 — 2 May 2004

“…Smirnov (2001) proved Cardy's formula for crossing probabilities of critical site percolation on the triangular lattice, and indicated how to achieve the full scaling limit. Schramm (2007) provides a survey of SLE and associated problems and conjectures.…”