Analogue of Cardy's formula on the 60-degree parallelogram: a modular approach

Abstract

The Cardy-Smirnov function of a four-marked planar domain encodes the conjectural scaling limit of the crossing probability for critical percolation. Cardy predicted a closed formula for this limit on the rectangle in the cases of both site and bond percolation, and Smirnov proved the formula together with conformal invariance for critical site percolation on the triangular lattice. Kleban and Zagier later showed that on the rectangle the function admits a modular interpretation: it is determined by a modular functional equation together with a mild analytic ansatz. We carry out the analogue on the $π/3$ parallelogram. We derive a closed conformal-map formula for the Cardy-Smirnov function as an incomplete beta integral, prove a closed modular formula realizing it as an integral of the modular form $η(τ)^2η(3τ)^2$, and establish a uniqueness theorem showing that a single functional equation, together with a $q$-expansion ansatz and a nondegeneracy condition, determines the function uniquely.