Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Description

For critical site percolation on the triangular lattice, the Cardy-Smirnov function encodes the probability that the scaling limit contains a left-to-right crossing in a four-marked planar domain; on other lattices, the existence of such a scaling limit remains open. Cardy predicted a closed formula on the rectangle, and Smirnov proved conformal invariance and Cardy's formula on the equilateral triangle, from which the rectangular case follows. 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 analogous program on the 60º parallelogram. We derive a closed conformal-map formula for the Cardy--Smirnov function as an incomplete beta integral, prove a closed modular formula expressing it as an integral of $\eta(\tau)^2\eta(3\tau)^2$, and establish a uniqueness theorem showing that a single functional equation together with a q-expansion ansatz determines the function uniquely. Time permitting, we then discuss further directions for research, including a problem in asymptotic combinatorics arising from representation theory, and another relating to the monstrous moonshine theory of Conway and Norton.