Conformal Invariance in Percolation, Self-Avoiding Walks, and Related Problems

Abstract

Over the years, problems like percolation and self-avoiding walks have provided important testing grounds for our understanding of the nature of the critical state. I describe some very recent ideas, as well as some older ones, which cast light both on these problems themselves and on the quantum field theories to which they correspond. These ideas come from conformal field theory, Coulomb gas mappings, and stochastic Loewner evolution.This talk is about 'geometric' critical phenomena. These are random spatial processes, where either (1) the probability distribution is determined by equilibrium statistical mechanics, and we ask questions about geometrical properties, or (2) the probability distribution is itself geometrical in nature. The simplest example of (1) is clustering in percolation (see Fig. 1), in which the probability distribution * Plenary talk given at

Conformal Restriction, Highest-Weight Representations and SLE

“…Also, in the spirit of the conclusion of Cardy's review paper [10] and as already confirmed by [3], the rigorous SLE approach should hopefully become useful and exploited within the theoretical physics community.…”

“…It is probably worthwhile to spend some lines outlining our perception of the history of this subject (see also the recent review paper by Cardy [10]): It has been recognized by physicists some decades ago that two-dimensional systems from statistical physics near their critical temperatures have some universal features. In particular, some quantities (correlation length for instance) obey universal power laws near the critical temperature, and the value of the (critical) exponent in fact depends only on the phenomenological features of the discrete system (for instance, it is the same for the same model, taken on different lattices).…”

Unparticle Physics

“…It is not clear to me what 2D results translate into four dimensions because the phase space in two dimensions is so constrained. But there are certainly consequences for condensed matter physics, where conformal structures do exist (see, for example, [11]). (ii) The connection between operator scaling dimension in a CFT and missing energy distributions was made for ordinary particles with integral scaling dimension in [12].…”

Is Critical 2D Percolation Universal?

“…Under some symmetry conditions on T , Russo-Seymour-Welsh theory ensures that at criticality, the probability of C δ (Ω, A, B, C, D) is bounded away from both 0 and 1 as δ goes to 0. Its limit was conjectured by Cardy (see [6]) using non-rigorous arguments from conformal field theory; actual convergence was proved, in the case of the triangular lattice (embedded in such a way that its faces are equilateral triangles), by Smirnov (see [15,3]). We defer the statement of the convergence to a later time.…”