Conformal Invariance and Percolation

The talk presented at ICMP 97 focused on the scaling limits of critical percolation models, and some other systems whose salient features can be described by collections of random lines. In the scaling limit we keep track of features seen on the macroscopic scale, in situations where the short--distance scale at which the system's basic variables are defined is taken to zero. Among the challenging questions are the construction of the limit, and the explanation of some of the...

We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic L\"owner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In part...

An exact formula is given for the probability that there exists a spanning cluster between opposite boundaries of an annulus, in the scaling limit of critical percolation. The entire distribution function for the number of distinct spanning clusters is also given. These results are found using Coulomb gas methods. Their forms are compared with the expectations of conformal field theory.

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the cont...

We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.

We conjecture an exact form for an universal ratio of four-point cluster connectivities in the critical two-dimensional $Q$-color Potts model. We also provide analogous results for the limit $Q\rightarrow 1$ that corresponds to percolation where the observable has a logarithmic singularity. Our conjectures are tested against Monte Carlo simulations showing excellent agreement for $Q=1,2,3$.

The logarithmic conformal field theory describing critical percolation is further explored using Watts' determination of the probability that there exists a cluster connecting both horizontal and vertical edges. The boundary condition changing operator which governs Watts' computation is identified with a primary field which does not fit naturally within the extended Kac table. Instead a "shifted" extended Kac table is shown to be relevant. Augmenting the previously known log...

The review is a brief description of the state of problems in percolation theory and their numerous applications, which are analyzed on base of interesting papers published in the last 15-20 years. At the submitted papers are studied both the cluster system of the physical body and its impact on the object in general, and adequate mathematical tools for description of critical phenomena too. Of special interest are the data, first, the point of phase transition of certain of ...

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around th...

We consider discrete random fractal surfaces with negative Hurst exponent $H<0$. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level $h$. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter ($H$) family of percolation models with long-range correlation in the site occupation. The level clusters percolate a...