The next meeting of the Mathematical Foundations of Informatics seminar will be held on March 9, 2023, at 3 p.m. (UTC+2) in Room 503 of the Institute of Mathematics and Informatics. A talk on

Introduction to the dimer model on the plane: scaling limit and conformal invariance

will be delivered by Dr. Mikhail Basok, visiting researcher under the PIKOM program.

Abstract: Dimer model is a classical model in planar statistical physics. Given a finite graph, the model is described as a probability distribution on the set of dimer covers (=perfect matchings) of the graph. In the case when the graph is planar each dimer cover is described with the so-called height function, which is a certain function on the faces of the graph.
In this talk we consider a particular setup when the graph is given as a subgraph of a square lattice on the plane and the distribution on the space of dimer covers is uniform. Given a simply-connected domain we consider a sequence of such graphs approximating this domain as the step of the lattice tends to zero and sample the dimer model on each of the graphs. Classical theorem of Kenyon asserts that under a certain local combinatorial conditions the sequence of the corresponding (random) height functions has a conformally invariant limit. Moreover, this limit is proven to be the Gaussian free field with Dirichlet boundary conditions in the initial domain.

We will discuss heuristics behind this theorem and the approтach to proving it via discrete complex analysis developed by Kenyon. If time permits, we will also discuss random loop ensembles arising from a pair of two independent dimer covers and its convergence to a conformally invariant limit proved in our joint work with Dmitry Chelkak.