# Crossing probabilities, their densities, and modular forms

# Crossing probabilities, their densities, and modular forms

A crossing probability is the probability of finding a critical cluster that touches specified boundary arcs. We consider formulas for the prototypical example, the horizontal crossing probability Ph, which gives the probability of connecting the left and right sides of a rectangle containing a system at the two-dimensional percolation point, as well as various related crossing probabilities and probability densities (that a point z is in a specified cluster) on the rectangle. The density formulas apply to a range of two-dimensional critical models, not just percolation. We will begin with an introduction to modular forms, and illustrate their interest in this context. Surprisingly, all the crossing formulas mentioned have modular properties, being either modular forms, second-order modular forms or (meromorphic Hermitian) Jacobi modular functions. This is unexpected because a rectangle lacks toroidal symmetry; the origin of this modular behavior is a mystery.