Mathematics Colloquia and Seminars

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Description

We motivate this talk by considering 'Random-Turn Hex,' a probabilistic variant of the classic two-player board game. In the scaling limit, the winning probability of this game converges to the crossing probability of critical percolation, a value governed by Cardy's Formula. Kleban and Zagier famously demonstrated that for rectangular domains, these probabilities possess deep modular properties. In this talk, we extend this analysis to the 60 degree parallelogram, the domain of the Hex board. By analyzing the monodromy of the Schwarz-Christoffel map, we show that the crossing probability on the rhombus is governed by a subgroup of the modular group. Our derivation takes us on a tour through conformal mapping, hypergeometric differential equations, and the theory of modular forms.