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