Cardy's formula does not hold on some 2D lattices for critical two-dimensional percolation

TL;DR

This paper demonstrates that Cardy's formula, which predicts crossing probabilities in 2D critical percolation, does not hold universally across all lattice types, challenging its assumed generality.

Contribution

The study provides counterexamples showing Cardy's formula fails on certain 2D lattices, revealing limitations in its applicability.

Findings

Cardy's formula does not hold for some 2D lattices

Counterexamples include specific periodic triangular and square lattices

Challenges the universality of conformal invariance in percolation

Abstract

The scaling limit of crossing probabilities is believed to satisfy a conformal mapping formula, called Cardy's formula, in two-dimensional percolation at the criticality. The formula has been confirmed to hold for site percolation on the equilateral triangular lattice. In this paper, we show that Cardy's formula could not hold for some two-dimensional triangular and square-type lattices, in particular for some periodic 2D graphs.