A crossing probability for critical percolation in two dimensions

TL;DR

This paper derives exact formulas for crossing probabilities in 2D critical percolation, extending previous results and confirming them with numerical data, thus deepening understanding of percolation theory.

Contribution

It provides an explicit formula for the crossing probability pi_{hv} in 2D critical percolation, extending Cardy's work on pi_h and validating it against numerical results.

Findings

Derived an exact formula for pi_{hv}

Confirmed the formula's accuracy with numerical data

Extended Cardy's boundary operator approach to new crossing probability

Abstract

Langlands et al. considered two crossing probabilities, pi_h and pi_{hv}, in their extensive numerical investigations of critical percolation in two dimensions. Cardy was able to find the exact form of pi_h by treating it as a correlation function of boundary operators in the Q goes to 1 limit of the Q state Potts model. We extend his results to find an analogous formula for pi_{hv} which compares very well with the numerical results.