# Events

### February 2018

26/02/2018

Guillaume Rémy: The Fyodorov-Bouchaud formula and Liouville conformal field theory 15:00-16:00

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

05/03/2018

Domenico Marinucci (Rome): The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics 15:00-16:00

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.

05/03/2018

Matthias Winkel (Oxford): Squared Bessel processes and Poisson-Dirichlet interval partition evolutions 16:00-17:00 KCL

05/03/2018

Andreas Eberle (Bonn): Couplings, metrics and contraction rates for Langevin diffusions 17:00-18:00

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).