London Probability at King's


November 2017

Mateusz Majka
London Probability Seminar
15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

December 2017

Anne Estrade: A test of Gaussianity based on the excursion sets of a random field
London Probability Seminar
15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20
In the present talk, we deal with a stationary isotropic random field $X:\R^d \to \R$ and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets. On the one hand, we study the properties of these level functionals under the hypothesis that the random field $X$ is Gaussian. In particular, we focus on the mapping that associates to any $u$ the expected Euler characteristic of the excursion set above level $u$. On the other hand, we study the same level functionals under alternative distributions of $X$, such as chi-square or shot noise. The talk is based on a joint work with Elena Di Bernardino (CNAM Paris) and José R. Leon (Universidad Central de Venezuela).

January 2018

Nathanael Berestycki: The dimer model on Riemann surfaces
London Probability Seminar
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.

February 2018

Guillaume Rémy: The Fyodorov-Bouchaud formula and Liouville conformal field theory
London Probability Seminar
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.

March 2018

June 2018

Workshop "Random Waves in Oxford"
Oxford University