Cedric Boutillier: Minimal bipartite dimers and maximal Riemann surfaces

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.