Julien Berestycki (Paris)
Branching processes with selection
Abstract: Let us consider a model in which particles move in space and branch independently of one another but on which a selection mechanism is acting to keep the population size essentially constant. Several such models have recently been investigated by physicists (Brunet, Derrida et al.).
Non-rigorous arguments led them to formulate several remarkable predictions for those systems. In particular they argued that the asymptotic genealogy of the particles should be a universal object: the Bolthausen-Sznitlan coalescent. I will give an overview of some of those conjectures as well as some recent related rigorous results. (Based on joint work with N.Berestycki and J. Schweinsberg).
Ivan Corwin (Microsoft Research and MIT)
Beyond the Gaussian Universality Class
Abstract: The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers,particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.
The master field on the plane
Abstract: The Yang-Mills field on the plane is a collection of random matrices indexed by the set of loops based at the origin on the plane.
These matrices belong to a fixed compact group, for example the unitary group U(N), and the Yang-Mills field can be thought of as a random unitary representation of the group of reduced rectifiable loops, the group operation being concatenation.
I will describe the large N limit of this random reprensentation, in particular the fact that it converges almost surely towards a deterministic limit.
This limit, which is one instance of what physicists call the master field, takes the form of a plain deterministic real-valued function on the set of loops. This function can be computed by a recursive algorithm based on the Makeenko-Migdal equations, which are a graphical expression in terms of the geometry of loops of the algebraic structure of freeness.
Max von Renesse (Berlin)
Hamiltonian Mechanics on Wasserstein Space and Quantum Fluid Models
Abstract: Otto's Riemannian Framework for the Wasserstein space of probablity measures allows not only for first order gradient flows but also for second order Hamiltonian ODEs. As a result we give a concise representation of the Schrödinger equation for wave functions as an instance of Newton's classical law of motion on Wasserstein space, the two representations being related by a natural sympelctic morphism.
Introducting friction leads to dissipative quantum fluid models such as the Quantum Navier Stokes equation, which was derived as a model for a tagged particle in a many body quantum system. (Partially based on joint works with A. Jüngel and P. Fuchs (Vienna)).
Timo Seppalainen (University of Wisconsin-Madison)
KPZ universality and directed polymers in random environments
Lecture 1. KPZ universality for interacting stochastic models
Abstract: KPZ equation, fluctuation exponents and one-point distribution, and KPZ universality. 1+1 dimensional exactly solvable directed polymer models. Contrast with Gaussian universality.
Lecture 2. Directed polymers in random environments
Abstract: Weak and strong disorder. Martingale techniques for proving a central limit theorem in high temperature and localization in the so-called strong coupling regime (low temperature or dimensions 1 and 2).
Abstract: Burke property or output theorem in exactly solvable 1+1 dimensional directed polymer models, first zero temperature and then positive temperature. Fluctuation exponents for the log-gamma polymer.
Lecture 4. Tropical RSK correspondence and directed polymers
Abstract: The Robinson-Schensted-Knuth correspondence from combinatorics and its tropical counterpart introduced by Kirillov. Connection with exactly solvable polymer models.
A conditioning principle for Galton-Watson trees
Abstract: We discuss the behaviour of a Galton-Watson tree conditioned on its martingale limit being small. We prove that it converges to the smallest possible tree, giving an example of entropic repulsion where the limit has no entropy. We also discuss the first branching time of the conditioned tree (which turns out to be almost deterministic) and the strength of the first branching. This is a joint work with N. Berestycki (Cambridge), N. Gantert (Munich), P. Moerters (Bath).
Hastings-Levitov aggregation and the Brownian web
Abstract: In 1998 Hastings and Levitov proposed a family of models for planar random growth in which clusters are represented as compositions of conformal mappings. This family includes physically occurring processes such as diffusion-limited aggregation, dielectric breakdown and the Eden model for biological cell growth. I shall describe the limits that result from small particle size and rapid aggregation in a special case of this model. In particular I shall show how the Brownian web arises in a limit of the fine scale branching structure that is present within the cluster. This is based on joint work with James Norris (Cambridge University).