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Planar bipartite dimer model : discrete holomorphicity and Gaussian Free Field

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Planar bipartite dimer model : discrete holomorphicity and Gaussian Free Field

A classical theorem due to Kasteleyn says that the partition function of a planar dimer model equals to the Pfaffian of a properly signed adjacency matrix of the graph. In 2000, Kenyon proved that the fluctuations of the associated height function in special (so-called Temperleyan) discrete approximations to a given planar domain on refining square grids converge to the Gaussian Free Field. The starting point of Kenyon’s argument is an interpretation of the Kasteleyn matrix as a discrete Cauchy-Riemann operator; one of the observations that brought discrete holomorphic functions to the focus of research on critical 2d lattice models during the following decade. However, the dimer model is known to be very sensitive to boundary conditions and such a straightforward interpretation fails for other types of discrete domains. One of the most classical examples of a more complicated behavior are the so-called Aztec diamonds: in this case, frozen/liquid zones appear and the height fluctuations in the liquid zone converge to a Gaussian field but the two-point function is not the standard Green function. The goal of this talk is to briefly review these classical results and – at the very end – to indicate recent developments on generalizations of the discrete complex analysis philosophy beyond “standard” setups.

- Séminaire informel de Probabilités et Statistiques

Détails :

Orateur / Oratrice : Dmitry Chelkak
Date : 4 janvier 2022
Horaire : 11h00 - 12h00
Lieu : ENS – salle W
Address: 45 rue d'Ulm