SCIENTIFIC PROGRAMS AND ACTIVITIES

June 10
3:00 p.m.
Stewart Library, Fields Institute

Joseph Najnudel, University of Toulouse
Mod-* convergence: a generalization of the convergence in law

The central limit theorem says that after normalization, the sum of independent random variables converges to a gaussian random variable. It is natural to try to refine this result. One possible way, introduced by Kowalski and Nikeghbali, is to look at the characteristic function and compare it to the characteristic function of a gaussian variable with suitable variance. More precisely, a sequence of random variables is said to converge in the mod-gaussian sence if the quotient of these characteristic functions converges to a limiting function. In this talk, we present and generalize several examples for which this convergence holds, and we characterize the possible limiting functions.

May 30
10:00 a.m.
Room 210, Fields Institute

David Gamarnik, MIT

We discuss algorithmic hardness of solving combinatorial optimization problems on sparse graphs by means of local algorithms. Recently a framework for local algorithms was proposed based on the concept of i.i.d. factors. In particular, it was conjectured that such an algorithm should exist for the problem of finding a largest independent set in a random regular graph. We disprove this conjecture by showing that no local algorithm is capable of producing an independent set larger that some multiplicative factor of the optimal. Our approach is based on a powerful clustering phenomena discovered by statistical physicists in the context of spin glass theory, and recently confirmed by rigorous methods. To the best of our knowledge, our result is the first direct application of the spin glass theory methods to the area of algorithmic hardness.

Joint work with Madhu Sudan (Microsoft Research)

Pierre Patie, Cornell University
Some spectral problems associated to non-self-adjoint self-similar semi-groups

We start this talk by showing a one-to-one correspondence between the class of invariant Lamperti-Feller semi-groups and a subset of negative definite functions. It turns out that these
non-self-adjoint semi-groups are closely related to positive self-similar Feller processes which were introduced by Lamperti in 72 and have been studied intensively recently. We proceed by showing the existence of an intertwining relationship between this class of semi-groups and the semi-group of a radial Ornstein-Uhlenbeck process, a self-adjoint diffusion. Exploiting this connection, we provide NSF conditions for the existence of discrete spectrum of this class of semi-groups. When the spectrum is purely discrete we discuss eigenvalues expansions by describing the sequence of eigenfunctions and co-eigenfunctions. We also explain why this spectral expansion is indeed possible on a space of functions which suffices for the description of the semi-groups but fails on the full Hilbert space generated by the invariant measures of the semi-groups.
This study is carried jointly with Mladen Savov (University of Reading, UK).

Apr. 1
4:30 p.m.
Stewart Library, Fields Institute

Arno Kuijlaars, Leuven
The Hermitian two matrix model with quartic potential

I will discuss eigenvalues of random matrices from the Hermitian two-matrix model with an even quartic potential. The mean limiting eigenvalue distribution is governed by a vector equilibrium problem for three measures with external fields and an upper constraint. Varying the parameters one observes phase transitions at the closing or opening of a gap in the limiting spectrum. The talk is based on joint work with Maurice Duits (Stockholm) and Man Yue Mo (Bristol).

Wednesday,
Mar. 27,
3:10pm
Room 210, Fields Institute

Tom Alberts, Caltech
Dimension Spectrum of SLE Boundary Collisions

In the range 4 < \kappa < 8, the intersection of the Schramm-Loewner Curve (one of the central objects in the theory of 2-D Conformally Invariant Systems) with the boundary of its domain is a random fractal set. After reviewing some previous results on the dimension and measure of this set, I will describe recent joint work with Ilia Binder and Fredrik Viklund that partitions this set of points according to the generalized "angle" at which the curve hits the boundary, and computes the Hausdorff dimension of each partition set. The Hausdorff dimension as a function of the angle is what we call the dimension spectrum.

Special Probability Day Monday, March 18, 2013
University of Toronto, St George (downtown) campus.
Morning talks are in Wilson Hall, WI523 map.

10:10 Peter Winkler, Darthmouth College: New Extremes for Random Walk on a Graph
11:10 Michael Damron, Princeton University :Geodesics and direction in first-passage percolation
12:30 Lunch, TBA
Afternoon talks are in Bahen, BA6183. This is inside the math department. See map.
3:10 Richard Kenyon, Brown University: On the Ising model
4:10 Alice Guionnet, MIT: About heavy tails matrices

Mar. 11
3:00 p.m.
Room 230, Fields Institute

Ruth Williams (UCSD)
Correlation of Intracellular Components due to Limited Processing Resources

A major challenge for systems biology is to deduce the molecular interactions that underlie correlations observed between concentrations of different intracellular components. Of particular interest is obtaining an understanding of such effects when biological pathways share common elements that are limited in capacity. Here we use stochastic models to explore the effect of limited processing resources on correlations when these resources are positioned downstream or upstream of the molecular species of interest. Specifically, we consider two situations where correlations in protein levels are the object of interest: (i) degradation of different proteins by a common protease, and (ii) translation of different mRNA transcripts by a limited pool of ribosomes. In developing and analyzing stochastic models for these systems, we use insights from the mathematical theory of multiclass queues which was originally developed to understand congestion effects in telecommunication, computer, manufacturing and business systems. In both models we observe a correlation resonance: correlations tend to have a peak slightly beyond the point where the systems transition from underloading to overloading of the processing resources, although the sign of the correlation is different in the two cases. As time permits, related experimental work will be described.

This presentation is based on joint work with current or former members of the UCSD Biodynamics lab and in particular with William H. Mather, Natalie A. Cookson, Tal Danino, Octavio Mondragon-Palomino, Jeff Hasty and Lev S. Tsimring.

Mar. 11
4:30 p.m.
Room 230, Fields Institute

Matthias Keller (Jena)
Absolutely continuous spectrum of Galton-Watson trees

We study the discrete Laplace operator on multi-type Galton-Watson trees. We are interested in the case where the distribution of the branching lies in a neighborhood of a deterministic one. These deterministic trees are called trees of finite cone type and their spectrum consists of finitely many bands of purely absolutely continuous spectrum.

So, whenever the distribution is not far from being deterministic and such that each vertex has at least one forward neighbor, the operators on the Galton-Watson trees inherit most of the absolutely continuous spectrum from the deterministic case.

Mar. 1
3:00 p.m.
Room BA 3008,
Bahen Centre

Charles Bordenave (Toulouse)
An entropy for unimodular trees

Gabor Elek has proved that any unimodular tree with bounded degrees is the Benjamini-Schramm local weak limit of a sequence of finite graphs. We may then look for a quantitative version of this theorem and try to compute the number of graphs of size n which are close to a given unimodular tree. To perform this, we will introduce a natural notion of entropy. We will deduce large deviations principles for Erdos Renyi graphs and uniform graphs with given degree distribution. This is a joint work with Pietro Caputo.

Feb. 25
4:30 p.m.
Stewart Library

Arnab Sen (Cambridge)
A new approach to the Brownian web

The Brownian web corresponds informally to starting coalescing Brownian motions from every space-time point in 1+1 dimensions. The standard Brownian web, as defined by Fontes, Isopi, Newman and Ravishankar, is the scaling limit of coalescing random walks as long as the third moment of the jump distribution is finite. The third moment condition is known to be also necessary for this convergence to hold. Inspired by the work of Schramm and Smirnov on the scaling limit of critical planar percolation, we provide a new state space and topology for the Brownian web. In particular, this allows us to derive an invariance principle for coalescing random walks under an optimal second moment condition. Our approach is sufficiently simple and general that we can prove similar invariance result for coalescing random walks on Sierpinski gasket with little extra work. This is the first such result where the limiting paths do not enjoy the non-crossing property.

Joint work with Nathanael Berestycki (Cambridge) and Christophe Garban (ENS Lyon).

Christian Sadel (UBC)
Absolutely continuous spectrum for the Anderson model on trees

Trees are essentially the only structures where the existence of absolutely continuous spectrum is known for the Anderson model (the adjacency operator plus random potential). I will present one approach based on supersymmetry.

Feb. 11
4:30 p.m.
Room 230

Mike Molloy (Toronto)
Frozen Vertices in Colourings of a Random Graph

Over the past decade, much of the work on random k-SAT, colourings of random graphs, and other random constraint satisfaction problems has focussed on some foundational unproven hypotheses that have arisen from statistical physics. Some of the most important such hypotheses concern the “clustering’ of the solutions. It is believed that if the problem density is su?ciently high then the solutions can be partitioned into clusters that are, in some sense, both well-connected and well-separated. Furthermore, the clusters contain a linear number of “frozen variables”, whose values are ?xed within a cluster. The density where such clusters arise corresponds to an algorithmic barrier, above which no algorithms have been proven to solve these problems.

We prove that frozen vertices do indeed arise for k-colourings of a random graph, when k is a su?ciently large constant, and we determine the exact density threshold at which they appear.

Omer Angel (UBC).
Half planar maps

We characterise all measures on half planar maps that satisfy a domain Markov property, and discuss some of their geometric properties. Joint work with Gourab Ray.

Balázs Szegedy (Toronto)
Couplings of probability spaces and related issues

Couplings of probability spaces are extensively used in probability theory. One can use them to study the properties of various random processes. We present a different view point where we represent various algebraic and combinatorial structures in the coupling space of abstract probability spaces. We demonstrate this direction in ergodic theory, higher order Fourier analysis, and combinatorics (Sidorenko's conjecture).

Dec. 3
4:30 pm

Daniel Remenik (Universidad de Chile)
Determinantal line ensembles

During the last decade there has been a lot of interest in certain stochastic processes which arise from families of non-intersecting paths. Prominent examples are the Dyson Brownian motion from random matrix theory and the Airy processes describing the spatial fluctuations of certain random growth models. The distribution of these processes are typically given by Fredholm determinants of what are known as "extended kernels". Recently, a second type of Fredholm determinant formula, in
terms of certain boundary value operators, has been very useful in studying properties of Airy processes. I will explain how this second type of formula holds in great generality and give examples of how it can be applied to many processes of interest. Furthermore, I will describe how this type of formulas can be obtained directly from the non-intersecting nature of the families of paths considered (through the Karlin-McGregor/Lindström-Gessel-Viennot formula).

This is joint work with A. Borodin and I. Corwin.

Nov. 26
4:30 pm

Michal Kotowski (Waterloo):
Random groups and property (T)

I will introduce the notions of random groups in Gromov model and property (T). Random groups are an important object of study and a source of interesting examples in geometric group theory. Then I will sketch the proof that for density d > 1/3 random groups have property (T) with high probability. The techniques used here come from spectral graph theory and random graphs. The talk is based on the paper http://arxiv.org/abs/1106.2242.

Nov. 12
4:30
Fields Institute, Room 210

Jeremy Quastel (University of Toronto)
How far does stuff get in an interacting system of asymmetric random walks on Z^d?

In a system of non-interacting asymmetric random walks on Z^d a typical particle's variance is order t. If the particles interact, the situation is very different. A single particle may be sub-diffusive, but actually what we really care about is the diffusivity of the bulk. This is supposed to scale universally according to the local structure of the flux, and the dimension. In lower dimensions it can be super-diffusive, with very precise conjectures coming from rough physical arguments. In joint work with Benedek Valko we give a proof of diffusivity/superdiffusivity in the various regimes for a large class of such lattice gas models.

Nov. 9 at 2:10 p.m.
GB 221

We discuss the answer to a question of Churchley who asked how long it will take a cop to catch a drunk robber who moves randomly. We begin by discussing other variants of the cop-robber paradigm. This is joint work with Alex Scott, Colin McDiarmid, and Ross Kang.

Nov. 7
4:10
BA6183, Bahen Center, 40 St George St

Of interest to the TPS: Math Department Colloquium

Ivan Corwin (Clay Mathematics Institute, MIT, Microsoft Research)

Over the last 15 years researchers in probability, integrable systems and mathematical physics have uncovered a few important sources of integrable probabilistic models which have allowed then to access and describe universal phenomena of certain classes of disordered systems. The purpose of today's talk is to identify one such universality class containing growth models, driven diffusive lattice gases and directed polymer models and explain how representation theory (in the form of symmetric functions) serves as a significant source of integrable probability.

Nov. 5
4:30

On Z^d the expected distance of a walker from its starting point after n steps is of order root n. On the free group (the regular tree), it is of order n. What exponents apart from 1/2 and 1 are achievable?
I will explain why 1/2 is the minimal exponent. Nothing between 1/2 and 3/4 is known. I will also present a construction, joint with Gidi Amir, that achieves every power between 3/4 and 1.

Oct. 1
4:30 p.m.

Viktor Harangi (University of Toronto)
Independent sets and the minimum eigenvalue in transitive graphs

Hoffman's theorem gives an upper bound on the independence ratio of regular graphs in terms of the minimum eigenvalue of the adjacency matrix. We use invariant Gaussian processes on graphs to get a lower bound in the vertex-transitive case. Joint work with Bálint Virág.