Cedric Boutillier: Minimal bipartite dimers and maximal Riemann surfaces

Speaker
Cedric Boutillier, Sorbonne Université

Abstract
In 2002, Richard Kenyon introduced critical (trigonometric) weights for
dimers on isoradial graphs, and gave an explicit expression for the
inverse Kasteleyn matrix, the fundamental tool to compute local
statistics for the dimer model. In this talk, we review results of a series of papers in collaboration with Béatrice de Tilière and David Cimasoni to extend Kenyon’s result to
a larger family of weights. For Kasteleyn matrices constructed from
theta functions on a maximal Riemann surface, with a formula due to
Vladimir Fock, we show that there is a two-dimensional family of
inverses with an explicit integral representation, which have a locality
property. We will discuss some applications of these results to the spectral
Kenyon-Okounkov’s spectral theorem, and to generalized integrable
Laplacians on isoradial graphs.