On Crossing Event Formulas in Critical Two-Dimensional Percolation

TL;DR

This paper investigates crossing formulas in critical 2D percolation, showing they simplify in equilateral triangle domains and relate to elliptic functions, which may aid rigorous proofs.

Contribution

It demonstrates the simplification of crossing formulas in equilateral triangles and connects them to elliptic functions, suggesting a focus on triangular lattices for proofs.

Findings

Crossing functions simplify in equilateral triangle domains.

Crossings can be expressed via equianharmonic elliptic functions.

Simplifications facilitate potential rigorous proofs.

Abstract

Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts's formula for the horizontal-vertical crossing probability and Cardy's new formula for the expected number of crossing clusters. It is shown that under the assumption of conformal invariance, they simplify when the spatial domain is taken to be the interior of an equilateral triangle. The two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry. This suggests that rigorous proofs of Watts's formula and Cardy's new formula will be easiest to construct if the underlying lattice is triangular. The simplification in a triangular domain of Schramm's `bulk Cardy's formula' is also studied.