Crossing Probabilities in Critical 2-D Percolation and Modular Forms

TL;DR

This paper investigates the modular properties of crossing probabilities in critical 2D percolation, revealing unusual modular behavior and the interplay between conformal and modular invariance, supported by numerical and theoretical analysis.

Contribution

It uncovers the vector modular form structure of crossing probability derivatives and explores their modular transformations, extending understanding of symmetries in 2D percolation.

Findings

Derivatives of crossing probabilities form a vector modular form.

Horizontal and horizontal-vertical crossing probabilities exhibit modular transformation properties.

Evidence of additional symmetries beyond conformal invariance in percolation.

Abstract

Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossing probabilities transform as a vector modular form, one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form. We include consideration of the interplay between conformal and modular invariance that arises.