On the Rate of Convergence for Critical Crossing Probabilities

TL;DR

This paper establishes a power law estimate for how quickly crossing probabilities in certain percolation models converge to Cardy's Formula, enhancing understanding of critical phenomena in statistical physics.

Contribution

It provides the first quantitative rate of convergence for crossing probabilities in models where Cardy's Formula is valid, extending previous qualitative results.

Findings

Power law estimate for convergence rate

Quantitative bounds on crossing probabilities

Extension to generalized percolation models

Abstract

For the site percolation model on the triangular lattice and certain generalizations for which Cardy's Formula has been established we acquire a power law estimate for the \emph{rate} of convergence of the crossing probabilities to Cardy's Formula.