A scaling limit of crossing probabilities for critical percolation on the square lattice

TL;DR

This paper investigates the scaling limit of crossing probabilities in critical percolation on the square lattice, demonstrating the existence of such a limit in an equilateral triangle and showing that Cardy's formula does not apply.

Contribution

It establishes the existence of a scaling limit for crossing probabilities and proves that Cardy's formula is invalid for critical percolation on the square lattice.

Findings

Scaling limit exists for crossing probabilities in critical percolation on the square lattice.

Cardy's formula does not hold for the square lattice at criticality.

Provides insights into geometric properties of percolation models.

Abstract

We show the existence of a scaling limit for the crossing probabilities on the square lattice in an equilateral triangle for the critical percolation. We also show that Cardy's formula does not hold on the square lattice for the critical percolation.