London Probability at King's


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

Domenico Marinucci (Rome): The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics
London Probability Seminar
We shall first review some recent results on the geometry of Lipschitz-Killing curvatures for the excursion sets of random 2-dimensional spherical harmonics f_{\ell} of high degree. We shall then focus on the asymptotic behaviour of the nodal length, i.e. the length of their zero set. It is found that the nodal lengths are asymptotically equivalent, in the L^2 -sense, to the "sample trispectrum", i.e., the integral of the fourth-order Hermite polynomial of the values of f_{\ell}. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit. Based on joint work with Maurizia Rossi and Igor Wigman.
Andreas Eberle (Bonn): Couplings, metrics and contraction rates for Langevin diffusions
London Probability Seminar
Carefully constructed Markovian couplings and specifically designed Kantorovich metrics can be used to derive relatively precise bounds on the distance between the laws of two Langevin processes. In the case of two overdamped Langevin diffusions with the same drift, the processes are coupled by reflection, and the metric is an L1 Wasserstein distance based on an appropriately chosen concave distance function. If the processes have different drifts, then the reflection coupling can be replaced by a „sticky coupling“ where the distance between the two copies is bounded from above by a one-dimensional diffusion process with a sticky boundary at 0. This new type of coupling leads to long-time stable bounds on the total variation distance between the two laws. Similarly, two kinetic Langevin processes can be coupled using a particular combination of a reflection and a synchronous coupling that is sticky on a hyperplane. Again, the coupling distance is contractive on average w.r.t. an appropriately designed Wasserstein distance. This can be applied to derive new bounds of kinetic order for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. Similar approaches are also useful when studying mean-field interacting particle systems, McKean-Vlasov diffusions, or diffusions on infinite dimensional state spaces. (joint work with Arnaud Guillin and Raphael Zimmer).

June 2018

Workshop "Random Waves in Oxford"
Oxford University