Stewart Library, Fields Institute
Joseph Najnudel, University of Toulouse
Mod-* convergence: a generalization of the convergence
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.
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)
Stewart Library, Fields Institute
Pierre Patie, Cornell University
Some spectral problems associated to non-self-adjoint self-similar
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
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).
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).
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.
Probability Day Monday, March 18, 2013
University of Toronto, St George (downtown) campus.
Morning talks are in Wilson Hall, WI523
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
4:10 Alice Guionnet, MIT: About heavy tails matrices
Room 230, Fields Institute
Ruth Williams (UCSD)
Correlation of Intracellular Components due to Limited
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.
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
Room BA 3008,
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.
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 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.
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
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.
Room BA 1180,
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
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
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
This is joint work with A. Borodin and I. Corwin.
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.
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
Nov. 9 at 2:10 p.m.
Reed (McGill University)
How long does it take to catch a drunk miscreant?
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.
BA6183, Bahen Center, 40 St George St
Of interest to the TPS: Math Department Colloquium
Ivan Corwin (Clay Mathematics Institute, MIT, Microsoft
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
Virag (University of Toronto)
How far is the random walker on a group?
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.
|No seminar October
due to Fields
Talagrand's Majorizing Measures theorem
Viktor Harangi (University of Toronto)
Independent sets and the minimum eigenvalue in transitive
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.