Conformal Invariance and Percolation

TL;DR

This paper introduces conformal field theory methods to analyze two-dimensional critical percolation, deriving probabilities of boundary-connected clusters and their expected counts without prior CFT knowledge.

Contribution

It provides an accessible introduction to applying conformal field theory to percolation, including new derivations of boundary cluster probabilities and counts.

Findings

Derived probability of boundary-connected clusters in 2D percolation

Calculated mean number of such clusters

Presented alternative approaches to these results

Abstract

These lectures give an introduction to the methods of conformal field theory as applied to deriving certain results in two-dimensional critical percolation: namely the probability that there exists at least one cluster connecting two disjoint segments of the boundary of a simply connected region; and the mean number of such clusters. No previous familiarity with conformal field theory is assumed, but in the course of the argument many of its important concepts are introduced in as simple a manner as possible. A brief account is also given of some recent alternative approaches to deriving these kinds of result.