# 2018-19

*The probability seminar takes place on Wednesdays at 4:00 pm in room B3.02.*

Organisers: Wei Wu, Stefan Adams, Stefan Grosskinsky

###### Term 1 2018-19

**October 3: TBA**

**October 10: Milton Jara (IMPA)**

**Title:** Entropy methods in Markov chains

**Abstract:** Building upon Yau's relative entropy method, we derive a new strategy to obtain scaling properties of Markov chains with a large number of components. As an application, we obtain very precise estimates on the mixing properties of a mean-field spin system. Time permitting, we will also discuss the derivation of the speed of convergence of the hydrodynamic limit and non-equilibrium fluctuations of interacting particle systems.

**October 17: Christian Webb (Aalto)**

**Title: **On the statistical behavior of the Riemann zeta function

**Abstract:** A notoriously difficult problem of analytic number theory is to describe the behavior of the Riemann zeta function on the critical line. After reviewing some basic facts about the zeta function, I will discuss what can be said if the problem is relaxed by considering the behavior of the zeta function in the vicinity of a random point on the critical line.

Time permitting, I will also discuss how this problem is related to various models

of probability theory and mathematical physics. The talk is based on joint work with Eero Saksman.

**October 24: TBA**

**October 31: Paul Dario (ENS Paris)**

**Title: **Quantitative result on the gradient field model through homogenization.

**Abstract: **Consider the standard uniformly elliptic gradient field model. It was proved by Funaki and Spohn in 1997, by a subadditivity argument, that the finite volume surface tension of this model converges to a limit called the surface tension. The goal of this talk is to show how one can use the tools developed in recent the theory of stochastic homogenization to obtain an algebraic rate of convergence of the surface tension. The analysis relies on the study of dual subadditive quantities, useful in stochastic homogenization, as well as a variational formulation of the partition function and the notion of displacement convexity from the theory of optimal transport.

**November 7: Alexandros Eskenazis (Princeton)**

**Title:** Nonpositive curvature is not coarsely universal

**Abstract: **A complete geodesic metric space of global nonpositive curvature in the sense of Alexandrov is called a Hadamard space. In this talk we will show that there exist metric spaces which do not admit a coarse embedding into any Hadamard space, thus answering a question of Gromov (1993). The main technical contribution of this work lies in the use of metric space valued martingales to derive the metric cotype 2 inequality with sharp scaling parameter for Hadamard spaces. The talk is based on joint work with M. Mendel and A. Naor.

**November 14: David Dereudre (Lille)**

**Title:** DLR equations and rigidity for the Sine-beta process

**Abstract:** We investigate Sine β, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of

one-dimensional log-gases, or β- ensembles, at inverse temperature β > 0. We adopt a statistical physics perspective, and give a description of

Sineβ using the Dobrushin-Landford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine β to a compact set,

conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration.

Moreover, we show that Sineβ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying

inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough

to include more general long range interactions in arbitrary dimension. (joint work with A. Hardy, M. Maïda and T. Leblé)

**November 21: Nadia Sidorova (UCL)**

**Title: **Localisation and delocalisation in the parabolic Anderson model****

**Abstract: **The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It describes the mean-field behaviour of a continuous-time branching random walk. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials it is localised at just one point. However, in a partially symmetric parabolic Anderson model, the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions.****

**November 28: Steffen Dereich (Münster)**

Title: Quasi-processes for branching Markov chains

**Abstract: **The concept of quasi-process stems from probabilistic potential theory. Although the notion may not be that familiar nowadays, it is connected to various current developments in probability. For instance, interlacements, as introduced by Sznitman, are Poisson point processes with the intensity measure being a quasi-process. Furthermore, extensions of Markov families as recently derived for certain self-similar Markov processes are closely related to quasi-processes and entrance boundaries. In this talk, we start with a basic introduction of parts of the classical potential theory and then focus on branching Markov chains. The main result will be a spine construction of a branching quasi-process.

The talk is based on joint work with Martin Maiwald (WWU Münster).

**December 5: Weijun Xu (Oxford)
**

**Title:** Weak universality of KPZ

**Abstract: **We establish weak universality of the KPZ equation for a class of continuous interface fluctuation models initiated by Hairer-Quastel, but now with general nonlinearities beyond polynomials. Joint work with Martin Hairer.