# Emerging Trends in Probability Theory

## Abstracts for the talks

**Louis-Pierre Arguin****Extreme values of correlated Gaussian fields: the spin glass perspective**

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.

**Vincent Beffara****Recent progress in 2D statistical physics**

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.

**Thierry Bodineau****Large deviations and non-equilibrium statistical mechanics**

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.

**Ivan Corwin****Integrable particle systems and Macdonald processes**

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.

**Alison M. Etheridge****Modelling Evolution**

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.

**Alan Hammond****Self-avoiding walk's endpoint displacement**

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.

**Antti Knowles****Random band matrices and the extended states conjecture**

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.

**Malwina Luczak****Epidemics on random graphs**

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.

**Grégory Miermont****Random maps and 2-dimensional random geometries**

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).

**Jeremy Quastel****The Kardar-Parisi-Zhang equation and universality class**

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.

**Fabio Toninelli****Random discrete interfaces and stochastic dynamics**

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.

**Vladimir Vovk****Game-theoretic probability: brief review**

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.

## Date and Location

**April 29 - May 03, 2013**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## 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**Jörg Lehnert**

Max-Planck-Institut für Mathematik in den Naturwissenschaften