Dynamics of Discrete Holomorphic Functions via Combinatorics

by Paul Melotti (Université Paris-Saclay)

Friday Nov 17, 2023, 2:00 PM → 4:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Probability and analysis informal seminar

There exists several ways to discretize holomorphic functions. One of them is based on Schramm's orthogonal circle patterns, and their generalization to so-called "cross-ratio maps" and "P-nets". These systems are naturally associated with a discrete time dynamics. I will mention results and open problems about this dynamics, in particular the "Devron" property, that states that singularities cannot be escaped by reversing time. I will show that these questions can be tackled by identifying those (birational) dynamics with the dSKP equation, which itself can be identified with partition functions of (oriented) dimers, a famously integrable model of statistical mechanics. Based on joint works with Niklas Affolter, Béatrice de Tilière, Jean-Baptiste Stiegler.

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