Critical Percolation in Finite Geometries

TL;DR

This paper uses conformal field theory to compute boundary crossing probabilities in 2D percolation, revealing invariance under conformal maps and confirming predictions with numerical data.

Contribution

It demonstrates that crossing probabilities at the percolation threshold are conformally invariant, extending the understanding of critical phenomena in finite geometries.

Findings

Crossing probabilities are invariant under conformal transformations.

Predictions for crossing probabilities in rectangles match numerical results.

Conformal invariance extends beyond scale invariance in critical percolation.

Abstract

The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.