The goal of this conference is to provide an overview of the major recent developments in probability theory and neighboring areas. Speakers are supposed to motivate their work and embed it into a general context, talks are supposed to be also accessible to non-probabilists. The format of the conference is supposed to be Oberwolfach-style, i. e. with sufficient time for questions and discussions.

Speakers

Louis-Pierre Arguin

Université de Montréal

Vincent Beffara

ENS de Lyon

Thierry Bodineau

Ecole Normale Supérieure

Ivan Corwin

Clay Mathematics Institute, MIT and Microsoft Research

A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting "continuum random surface".
In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.
Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

Two-dimensional models of statistical physics have long been studied by physicists, using tools such as quantum and conformal field theories and renormalization groups as well as through explicit computations in integrable cases. On the mathematics front, two objects were introduced over the last decade, shedding new light on their geometry: first, stochastic Loewner evolutions, proved by Schramm to be the unique possible scaling limits of models exhibiting conformal invariance; second, (para)fermionic observables, used by Smirnov to actually prove conformal invariance of several of them. I will present a panorama of these recent advances and some of the most puzzling open questions remaining to be solved.

Many interesting particle systems are described by stochastic dynamics without the basis of a known underlying Hamiltonian. Their stationary states will be in general non-equilibrium stationary states which cannot be easily computed as there is currently no counterpart to the Gibbs theory for equilibrium systems. Thus a challenging issue would be to provide a probabilistic description of the non-equilibrium states and describe their limiting structure when the number of particles diverges.
In this talk, we will review some results on the steady states of diffusive systems maintained off equilibrium by two heat baths at unequal temperatures. Using the framework of the hydrodynamic limits, we will discuss the large deviations of the heat current through these systems. In particular, we will explain the occurrence of a dynamical phase transition which may occur for some models.

Random discrete interfaces are a classical object in statistical mechanics: they provide for instance effective models for phase separation boundaries in spin systems. Similar objects arise also naturally in combinatorics: for instance, discrete random surfaces are associated naturally to random dimer coverings of bipartite graphs.
In recent years, a lot of activity has focused on studying equilibrium statistical properties of random surfaces, notably, proving convergence to the so-called "gaussian free field" for (2+1)-dimensional surfaces.
An equally interesting but mathematically much less developed topic is that of studying stochastic (Markov) dynamics of discrete interfaces. Motivations arise both from statistical physics (understanding the time evolution of phase boundaries) and from combinatorics/theoretical computer science (Markov Chain Monte Carlo algorithms providing an efficient way of counting and of uniformly sampling dimer coverings). In the "diffusive limit" where space and time are suitably rescaled, the stochastic interface dynamics is in many cases believed to converge to a deterministic evolution of mean curvature type. We will discuss some recent developments, the related mathemaical difficulties and some perspectives.

The study of extremal values of a large collection of random variables dates back to the early 20th century and has been well established in the case of independent or weakly correlated variables. On the other hand, few universality results are known in the case where the random variables are strongly correlated. In the early 1980's, statistical physicists (with notable contributions by Giorgio Parisi) have proposed a compelling universal picture to understand the extremal values of correlated variables for a broad class of models. This picture was largely inspired from the study of spin glass models in physics. In this talk, I will describe the statistical physics approach in the case of Gaussian fields and survey recent rigorous results in establishing this picture. A particular emphasis will be put on the Sherrington-Kirkpatrick model and log-correlated Gaussian fields such as branching Brownian motion and the 2D Gaussian free field.

Random matrices were introduced in the 80s to model disordered quantum systems on large graphs (typically lattices). They provide a means of interpolating between random Schrodinger operators and mean-field models such as Wigner matrices. On the one-dimensional lattice it is conjectured that as one increases the band width a sharp transition occurs from the localized to the delocalized regime. In parallel, the local spectral statistics undergo a transition from Poisson to random matrix statistics.
I give an overview of recent progress in understanding the eigenvector and eigenvalue distribution of random band matrices. I mainly focus on the derivation of delocalization bounds on the eigenvectors. I outline two approaches: one based on perturbative renormalization and the other on the averaging of fluctuations among resolvent entries.

The endpoint of n-step self-avoiding walk in Z^d is predicted to have a typical distance from the origin of the order of n^{3/4} when d=2; this distance is numerically determined to be of the order of n^{0.59...} when d=3. In work with Hugo Duminil-Copin, and more recently also with Alexander Glazman and Ioan Manolescu, we have rigorous results excluding the extremes of fast and slow behaviour for endpoint displacement: ballisticity and localization near the origin. The talk will outline some of the main arguments used in establishing these two assertions.

In recent years, various models of random graphs beyond the standard G(n,p) random graph have been extensively studied. In particular, there has been increasing interest in modelling epidemics on random graphs.
In this work, we study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In the SIR model, infective vertices infect each of their susceptible neighbours, and recover, at a constant rate.
Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only few infections occur. Above the threshold a large outbreak may occur. Conditional on that, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time.
This is joint work with Svante Janson and Peter Windridge.

The standard approach to probabilistic modelling is to assume a probability measure generating the observed outcomes. Game-theoretic probability weakens this assumption but still allows one to obtain many familiar results, such as laws of large numbers and iterated logarithm, central limit theorems, large deviation inequalities, and zero-one laws. It also leads to completely new results.

This will be a survey talk on recent progress on the size and distribution of fluctuations for models in the 1+1 dimensional KPZ universality class, in particular, the KPZ equation itself.

A large class of one dimensional particle systems are predicted to share the same universal long-time/large-scale behaviors. By studying certain integrable models within this (Kardar-Parisi-Zhang) universality class we access what should be universal statistics and phenomena. The purpose of today's talk is to explain how representation theory (in the form of symmetric function theory) is the source of integrability within this class. We develop the theory of Macdonald processes (generalizing Okounkov and Reshetikhin's Schur processes) which unites integrability in various areas of probability including directed polymers, particle systems, growth processes and random matrix theory. We likewise develop the many body system approach to integrable particle systems.

The basic challenge of mathematical population genetics is to understand the relative importance of the different forces of evolution in shaping the genetic diversity that we see in the world around us. This is a problem that has been around for a century, and a great deal is known. However, a proper understanding of the role of a population's spatial structure is missing. Recently we introduced a new framework for modelling populations that evolve in a spatial continuum. In this talk we briefly describe this framework before outlining some preliminary results on the importance of spatial structure for natural selection.

Participants

Louis-Pierre Arguin

Université de Montréal

Scott Armstrong

Universite Paris - Dauphine

Vincent Beffara

ENS de Lyon

Thierry Bodineau

Ecole Normale Supérieure

Erwin Bolthausen

University of Zurich

Ljudmila Bordag

Hochschule Zittau/Goerlitz

Mireille Bousquet-Mélou

CNRS, Université Bordeaux 1

Ivan Corwin

Clay Mathematics Institute, MIT and Microsoft Research

Frank den Hollander

Universiteit Leiden

Alison M. Etheridge

University of Oxford

Gero Friesecke

Technische Universität München

Peter Friz

TU and WIAS Berlin

Martin Hairer

University of Warwick

Alan Hammond

University of Oxford

Martina Hofmanova

ENS de Cachan Bretagne

Venera Khoromskaya

Max-Planck Institute for Mathematics in the Sciences

Boris Khoromskij

MPI MiS

Antti Knowles

Courant Institute of Mathematical Sciences.

Malwina Luczak

Queen Mary, University of London

Daniel Marahrens

MPI MIS Leipzig

Grégory Miermont

Université Paris-Sud

Eckehard Olbrich

Max Planck Institute for Mathematics in the Sciences

Oliver Pfante

MPI MIS

Jeremy Quastel

University of Toronto

Michael Röckner

Universität Bielefeld

Artem Sapozhnikov

MPI Leipzig

Herbert Spohn

Technische Universität München

Sven Stodtmann

BASF SE & Universität Heidelberg

Lorenzo Taggi

Max Planck Institute for Mathematics in the Sciences

Fabio Toninelli

University Lyon 1

Tat Dat Tran

Max Planck Institute for Mathematics in the Sciences

Max von Renesse

Universität Leipzig

Vladimir Vovk

Royal Holloway, University of London

Julian Wergieluk

TU Chemnitz

Ivan Yamshchikov

Hochschule Zittau-Goerlitz

Ofer Zeitouni

Weizmann Institute of Science

Scientific Organizers

Wolfgang Hackbusch

Max Planck Institute for Mathematics in the Sciences

Jürgen Jost

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Felix Otto

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Erwin Bolthausen

Universität Zürich

Administrative Contact

Katja Heid

Max Planck Institute for Mathematics in the Sciences
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Jörg Lehnert

Max-Planck-Institut für Mathematik in den Naturwissenschaften
Contact via Mail