Cardy's Formula for Certain Models of the Bond-Triangular Type

TL;DR

This paper introduces a new class of 2D percolation models with local correlations, demonstrating that they exhibit critical behavior and satisfy Cardy's formula in the continuum limit, extending Smirnov's results.

Contribution

It extends Cardy's formula applicability to a non-trivial class of correlated 2D percolation models, broadening understanding of critical phenomena.

Findings

Model exhibits typical critical behavior.

Cardy-Carleson functions satisfy Cardy's formula.

Extends Smirnov's results to new models.

Abstract

We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indicative of critical behavior in 2D percolation problems. Subsequently, the so called Cardy-Carleson functions are demonstrated to satisfy, in the continuum limit, Cardy's formula for crossing probabilities. This extends the results of S. Smirnov to a non-trivial class of critical 2D percolation systems.