Video versions of all lectures and talks are available at en.chebyshev.spb.ru and lektorium.tv
Preliminary abstracts ## Longer courses## Alexei Borodin: Integrable models of random growth and branching graphsThe goal of the mini-course is to explain how representation theory helps to discover and analyze integrable structures in probability. The relevant representation theoretic constructions involve representations and characters of symmetric and unitary groups and their infinite-dimensional analogs, Schur and Macdonald symmetric polynomials, and Whittaker eigenfunctions of the quantum Toda lattice. The probabilistic counterparts are random growth models in one and two space dimensions, last passage percolation and directed polymers, solutions of the stochastic heat equation and Kardar-Parisi-Zhang equation and associated quantum many-body systems.Video 1, Video 2 ## Yuval Peres: Markov chains: mixing times, hitting times, and cover timesOne aim of the course is to describe the close connections, some old and some new, between three basic parameters of a Markov chain: Mixing time, hitting time and cover time. In particular, following classical results of Aldous, recent work ([2] and independently, [3]) shows that mixing time is equivalent to the maximal hitting time of large sets. Another aim is to present some useful methods for analyzing a Markov chain, namely coupling, stationary stopping times, optional stopping for suitable martingales, and the spectrum. Key examples will include random walks on the symmetric group (card shuffling), Glauber dynamics for the Ising model, and lamplighter groups. We will also describe the connection [6] of the cover time to the maximum of the Gaussian Free Field. Open problems we will explore are (1) Diaconis’ challenge to understand which chains exhibit cutoff (an abrupt change of the total variation distance to stationarity from near 1 to near zero); (2) for Glauber dynamics, when do extra updates assist mixing, and how do random updates compare with systematic updates? (3) On a Cayley graph, for how long is the rate of escape at least the square root of t/d,where t denotes time and d is the degree? (4) On a transitive graph, is mixing time always bounded above by the diameter squared times the degree? (5) In what generality is the diameter squared divided by log(n) a lower bound for the mixing time on an n-vertex graph? References: ## Ofer Zeitouni: Branching random walks and maxima of Gaussian free fieldsThe model of branching random walks, and its close relative branching Brownian motion, describes the evolution of particles that undergo both random motion (diffusion) and branching. The analysis employs a mixture of analytical tools (the KPP equation, recursions, monotonicity) as well as probabilistic ones (changes of measures, first and second moment methods, large deviations), especially in dimension 1, to which this mini-course will be devoted. The classical main results are due to Fisher and Kolmogorov-Petrovsky-Piscounov, as well as McKean and Bramson. There has been much recent progress concerning fine properties of the
leading particles, as well as in exploiting the link with
other problems involving random walk, and in particular cover time. Of
particular interest is the link with a class of Gaussian fields
called the (discrete) Gaussian free field, in dimension 2.
These lectures will present an introduction to both Branching random
walks and Gaussian free fields. A rough plan is the following:
1) Definition of branching random walks, law of large numbers,
recursions, tightness of maxima via the Dekking and Host argument,
first and second moments methods, location of maximum. Introduction to
the genealogy of the front. Inhomogeneous
media and phase transitions.
2) Rough tree structures and cover times.
3) Gaussian free fields on graphs: definitions, fluctuations of
maxima, basic inequalities (Borell-Tsirelson, Sudakov-Fernique).
Subsequences and tightness: the Dekking-Host argument revisited. An
introduction to the Dynkin isomorphism theorem.
4) Tightness of maximum for the 2D Gaussian free field: link with
modified branching random walks, fluctuations.
## Mini-courses## Yuri Bakhtin: Ergodic theory of the Burgers equation with random forceThe Burgers equation is one of the basic hydrodynamic models that describes the evolution of velocity fields of sticky dust particles. When supplied with random forcing it turns into an infinite-dimensional random dynamical system that has been studied since late 1990's. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by random force potential. Therefore, this is essentially a random media problem. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood well. However, recently a lot of progress was achieved for noncompact cases with forcing based on Poissonian point field. The course is naturally split into three parts. In the first part I
will recall basics on the Burgers equation and explain the central
ergodic result, so called One Force --- One Solution (1F1S) Principle,
for Burgers dynamics with random force in the compact case. The second
part will be devoted to localization and 1F1S in the quasi-compact
case where the random forcing decays to zero at infinity. In the third
part I will talk about 1F1S for the case of the forcing that is
stationary in space-time and has no decay at infinity. This part is
based on techniques that have been developed for last-passage
percolation and related models. It involves Kesten's concentration
inequality, greedy lattice animals, and other tools of modern
probability theory.
Video 1, Video 2 ## Vincent Beffara & Hugo Duminil-Copin: The 2D random-cluster model at and around criticalityThe aim of this mini-course is to present in some (possibly all) detail our recent results on the two-dimensional random-cluster model on the square lattice, namely the determination of the critical point and the sharpness of the phase transition. The course will be split into two roughly independent parts, the first one deriving $p_c$ through the use of sharp-threshold results and RSW-like estimates, and the second one exploiting Smirnov's fermionic observable away from the self-dual point to gain estimates on two-point functions.Video 1, Video 2 ## Sourav Chatterjee: Topics in concentration of measureClassical concentration of measure gives powerful general techniques for bounding fluctuations of complicated random objects. However, these bounds do not always match the correct order of fluctuations. In more and more modern examples coming from a variety of areas including random matrices, polymer models, spin glasses, etc, this sub-optimality of the classical bounds is turning out to be more of a rule than an exception. We will discuss how this deficiency of the classical theory has deep implications about the structure of the problems and some directions towards building a new theory.Video 1, Video 2 ## Alice Guionnet: Random matrices and the Potts model on random graphs(Slides) Matrix integrals have long been used in physics to study properties of statistical models on random graphs. In this mini-course, we will analyze this relation in the case of the particular model of the Potts model on random planar maps. Planar maps are graphs which can be properly embedded into the sphere. The Potts model on a given graph is a model of random Q-coloring of the set of vertices with a build-in interaction such that two neighboring vertices prefer to have the same color. The underlying graph will be chosen randomly in the set of planar maps. We will show that such a model is equivalent to some loop model on random graphs, where the random loops can be thought as the random boundary of regions of equal color. We will prove that such loop models can be represented by certain matrix integrals that can be easier to analyze (and compute) than the initial model itself. All the above notions will be introduced along the course and no prerequisite will be needed. ## Greg Lawler: An introduction to the Schramm-Loewner Evolution
The Schramm-Loewner evolution as orginally defined by Oded Schramm is a conformally invariant family of probability of measures connecting points on simply connected domains. I will give an introduction
taking a "partition function" approach to these curves
where the measures need not be probability measures. The
first lecture will be a survey of what is known while the second will concentrate on the mathematical tools to
prove results. |