08/01/2018Nathanael Berestycki: The dimer model on Riemann surfaces
The dimer model on a finite bipartite graph is a uniformly chosen perfect matching, i.e., a set of edges which cover every vertex exactly once. It is a classical model of mathematical physics, going back to work of Kasteleyn and Temeperley/Fisher in the 1960s, with connections to many topics including determinantal processes, random matrix theory, algebraic combinatorics, discrete complex analysis, etc. A central object for the dimer model is a notion of height function introduced by Thurston, which turns the dimer model into a random discrete surface. I will discuss a series of recent results with Benoit Laslier (Paris) and Gourab Ray (Victoria) where we establish the convergence of the height function to a scaling limit in a variety of situations. This includes simply connected domains of the plane with arbitrary linear boundary conditions for the height, in which case the limit is the Gaussian free field, and Temperleyan graphs drawn on Riemannsurfaces. In all these cases the scaling limit is universal (i.e., independent of the details of the graph) and conformally invariant. A key new idea in our approach is to exploit "imaginary geometry" couplings between the Gaussian free field and Schramm's celebrated SLE curves.
26/02/2018Guillaume Rémy: The Fyodorov-Bouchaud formula and Liouville conformal field theory
Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF. In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC. In this talk we will give a rigorous proof of this formula. Our method is inspired by the technology developed by Kupiainen, Rhodes and Vargas to derive the DOZZ formula in the context of Liouville conformal field theory on the Riemann sphere. In our case the key observation is that the negative moments of the total mass of GMC on the circle determine its law and are equal to one-point correlation functions of Liouville theory in the unit disk. Finally we will discuss applications in random matrix theory, asymptotics of the maximum of the GFF, and tail expansions of GMC.