Courses and talks

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 graphs

The 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 times

One 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:
[1] Markov Chains and Mixing Times, (2008). David A. Levin, Yuval Peres and Elizabeth L. Wilmer. Published by the Amer. Math. Soc.,
See http://research.microsoft.com/en-us/um/people/peres/markovmixing.pdf (errata are at http://pages.uoregon.edu/dlevin/MARKOV/errata.pdf)
[2] Mixing times are hitting times of large sets. Yuval Peres, Perla Sousi http://arxiv.org/abs/1108.0133
[3] Mixing and hitting times for finite Markov chains. Roberto Imbuzeiro Oliveira. http://front.math.ucdavis.edu/1108.1708
[4] Can extra updates delay mixing? Yuval Peres, Peter Winkler http://arxiv.org/abs/1112.0603
[5] Harmonic maps on amenable groups and a diffusive lower bound for random walks (2009). James R. Lee, Yuval Peres  Ann. Probability, to appear. http://arxiv.org/abs/0911.0274
[6] Cover times, blanket times, and majorizing measures (2010)
Jian Ding, James R. Lee, Yuval Peres http://arxiv.org/abs/1004.4371

Ofer Zeitouni: Branching random walks and maxima of Gaussian free fields

(You can find lecture notes here.)

The 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.
 5) Discussion of open problems.

Video 1, Video 2
   

Mini-courses

Yuri Bakhtin: Ergodic theory of the Burgers equation with random force

The 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 criticality

The 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 measure

Classical 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.

Video 1, Video 2


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.

Video 1, Video 2

Grigori Olshanski: Non-colliding processes with infinitely many particles

Models of Markov dynamics for N interacting non-colliding particles (N=1,2,...) have been studied in the Random Matrix literature since Dyson's paper (1962) on the matrix-valued Brownian motion. However, extension of the theory to the case of infinitely many particles presents substantial difficulties. For instance, one wants to take an appropriate large-N limit, but it is difficult to justify it, to prove that the Markov property is not destroyed, or to make sense of a formal expression for the hypothetical infinite-particle generator. I will describe a new and relatively elementary method of constructing models of infinite-dimensional Markov dynamics based on some ideas from representation theory of infinite-dimensional groups.

Video 1, Video 2

Mikhail Sodin: Random nodal portraits

We describe the progress and challenges of understanding the zero sets of smooth Gaussian random functions of several real variables. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution. 

We start with an intriguing heuristics suggested by Bogomolny and Schmit, which relates nodal portraits of 2D Gaussian monochromatic waves to bond percolation on the square lattice. Then we explain how tools from classical analysis (Gaussian isoperimetric inequality proved by Sudakov-Tsirelson and Borell, and Wiener's ergodic theorem) help to find the order of growth of the typical number of connected components of the zero set. This might be thought as a statistical version of Hilbert's 16th problem. 

The lectures are based on joint works with Fedor Nazarov.

Video 1, Video 2


Talks

Greg Lawler: Self-avoiding random walks

I will give a survey talk about two models: the self-avoidng walk and the loop-erased random walk and in doing so will also discuss the problem of simple random walk intersections. The behavior of the walks depends highly on the dimension of the walks. These problems have been a major focus of my research for many years (in some sense, for my entire research career), and I will state what is conjectured, what is known now rigorously, and what still remains as major open questions.

Video 1, Video 2

Yves Le Jan: Markov loops ensembles

Any semigroup of Markovian transition kernels defines a sigma-finite measure on loops which generates an increasing family of Poissonian loop ensembles. One can establish fundamental relations between these ensembles and several types of stochastic processes: Occupation fields, Gaussian free fields and their Wick powers, determinantal processes, branching forests, conformal loop ensembles and coalescence processes.

Video 1, Video 2

Anatoly Vershik: Combinatorial, probabilistic and operator problems of asymptotic theory of the representations

I will mention several problems which seems to be central from point of view more that 30 years distance of the theory. Among them:

1.Construction of representations and models of statistical physics;
2.Central (invariant) measures and traces of algebras;
3.Three dimension Young diagram;
4.Continual models and new infinite dimensional groups;
5.Statistics on Young diagrams and partitions.

Video 1, Video 2

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