Béatrice de Tilière (CEREMADE, University Paris-Dauphine PSL)
Title: The dimer model, an introduction. The dimer model on minimal graphs, the elliptic case and beyond
Abstract:
The dimer model represents the adsorption of diatomic molecules on the surface of a crystal. It is modeled through perfect matchings of a planar graph chosen with respect to the Boltzmann measure. When the graph is periodic and bipartite, Kenyon, Okounkov and Sheffield proved that the phase diagram is given by the spectral curve of the model, which has the remarkable property of being a Harnack curve. The first lecture will be an introduction to the dimer model, giving definitions, the founding result of Kasteleyn/Temperley-Fisher computing the partition function, and explaining the phase diagram obtained by Kenyon, Okounkov and Sheffield.
Another important result is the explicit local formula proved by Kenyon for the maximal entropy Gibbs measure when the underlying graph is isoradial and the model is critical. In a series of papers with Cédric Boutillier (Sorbonne University) and David Cimasoni (University of Geneva), we extend this work to a unified framework. We consider the model on minimal graphs and prove an explicit correspondence with the set of Harnack curves; we also prove local formulas for the two-parameter family of Gibbs measures. The second lecture will be dedicated to explaining these more recent works.
Alexandre Gaudillière (CNRS, University Aix-Marseille)
Title: Kirchhoff forests and applications
Abstract:
Kirchhoff forests are a variation on the uniform spanning tree theme. They constitute an integrable system that is intimately linked with the spectral properties of the Laplacian associated with the graph they cover. Since they are also efficiently sampled with a celebrated Abelian algorithm due to Wilson, they provide an effective tool to perform multiscale analysis or monitor branching processes and infection spreading within any finite network, or to produce quick spectrum estimates of any large real symmetric matrix.
After presenting this random spanning and rooted forest family by starting from the description of Wilson's algorithm, and after giving insights for this wide range of applications, we will present the «coupled forests process» and see how to use it for spectral estimation.
This is based on a joint and continuous work with Luca Avena, Fabienne Castell, Clothilde Mélot, Dominique Benielli, Nicolas Tremblay, Simon Barthelmé, Matteo Quattropani, Pierre-Olivier Amblard, Irina Gurewitsch, Twan Koperberg, Adoré Randriamandroso, Alessio Troiani and Sacha Duvergé.