January 7, 2014
Biased random walks on random graphs
Alexander Fribergh, CNRS Toulouse
We will focus on recent results in the field of random walks in random environments. The aim of this field is to study the long time behavior of a particle moving in random landscape.
One aspect of random walks in random environments that has attracted a lot of attention in recent years is to understand models in which we observe trapping phenomena. In those models, small pockets in the random landscape have a big impact on the behavior of the walk. This translates into a host of anomalous results for the long time behavior of the walk.
Biased random walks on random graphs is a natural model to observe trapping. Our main focus will be to present results on biased random walks on random trees (Galton-Watson trees) and in a random labyrinth (percolation cluster).
January 21, 2014
Intertwinings, wave equations and growth models
Mykhaylo Shkolnikov, UC Berkeley
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.
January 28, 2014
Self-avoiding walks, phase separation and KPZ universality
Alan Hammond, UC Oxford
A fundamental notion in statistical mechanics is phase transition: a microscopic system composed of a huge number of random particles depends on a thermodynamic parameter, and the system undergoes sudden changes in its large-scale structure as this parameter varies across a critical point.
Self-avoiding walk was introduced in the 1940s as a model in chemistry of a long chain of molecules, and is now viewed as a fundamental model in the rigorous theory of statistical mechancs. By introducing a positive parameter which provides a penalty to self-avoiding walk which is exponential in the walk's length, we obtain an example of phase transition. Recent work with Hugo Duminil-Copin shows that uniformly chosen self-avoiding walks of given high length move sub-ballistically, and this is related to the nature of this phase transition at the critical point. I will give an overview of the main elements of the proof.
Considering instead subcritical self-avoiding walk, and focussing on the planar case, we obtain a natural model for the problem of phase separation: when one substance is suspended in another, such as oil in water, a droplet forms, whose boundary approximates a smooth profile predicted by Wulff. Modelling the problem using a planar model such as subcritical self-avoiding walk, the fluctuation of the droplet boundary from its typical profile exhibits characteristic scaling exponents - 2/3 longitudinally and 1/3 latitudinally - which I derived a couple of years ago. The behaviour arises from a competition of local Gaussian randomness and global curvature constraints.
Phase separation in this guise is a static model. However, the Gaussian competition with curvature, and the two exponents, are shared by many dynamic models, of interfaces growing at random and subject to forces of surface tension. These models form the Kardar-Parisi-Zhang universality class. Resampling techniques from the phase separation papers find counterparts in more recent work, joint with Ivan Corwin, in which a natural Gibbs property is proved for the multi-line Airy process, which is a fundamental scaling limit encountered in KPZ universality. This Brownian-Gibbs property is valuable in, for example, improving regularity assertions about the Airy process. These ideas form the subject of the final part of the talk..
February 13, 2014
Long time behavior of ergodic Piecewise Deterministic Markov Processes. Some examples.
Hélène Guérin, Rennes et CRM
The Piecewise Deterministic Markov Processes (PDMPs) were introduced in the literature by Davis (1984, 1993) as a general class of non diffusion stochastic models. PDMPs are a family of Markov processes involving deterministic motion punctuated by random jumps. We will consider some ergodic PDMPs and study the speed of convergence of such processes to their invariant measure. This talk will focus on some (quite simple) examples to describe the variety of difficulties which can appear in such studies. Some recent results on this subject will be presented.
February 20, 2014
Poisson-Dirichlet statistics for the extremes of log-correlated Gaussian fields
Olivier Zindy, Paris 6
Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will describe a general approach based on rigorous works in spin glass theory to describe features of the Gibbs measure of these Gaussian fields. I will focus on a model defined on the periodic interval [0;1]. At low temperature, we show that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. This is joint work with Louis-Pierre Arguin
March 27, 2014
Strict Convexity of the Parisi Functional
Antonio Auffinger, Chicago
Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.
April 3, 2014
High-dimensional asymptotics for percolation of Gaussian free field level sets
Alex Drewitz, Columbia
We consider the Gaussian free field in $\Z^d,$ $d \ge 3.$ It is known that there exists a non-trivial phase transition for
its level set percolation; i.e., there exists a critical parameter $h_*(d) \in [0,\infty)$
such that for $h < h_*(d)$ the excursion set above level $h$ does have a unique infinite connected component, whereas for
$h > h_*(d)$ it consists of finite connected components only.
We investigate the asymptotic behavior of $h_*(d)$ as $d \to \infty$ and give some ideas on the proof of this asymptotics.
(Joint work with P.-F. Rodriguez)
April 17, 2014
Universality for the Stochastic Airy Operator
Brian Rider, Temple University
The Stochastic Airy Operator first arose as the continuum limit of certain (generalization of) ensembles of symmetric Gaussian random matrices in the vicinity of their spectral edge. We show that this picture persists for the general logarithmic gas on the line with uniformly convex polynomial potential. Based on joint work with Manjunath Krishnapur (IISC) and Bálint Virág (University of Toronto).
May 22, 2014
Andreas Kyprianou, Bath