
Toronto Probability Seminar 200910
held at the Fields Institute
Organizers
Bálint
Virág , Benedek
Valkó
University of Toronto, Mathematics and Statistics
For questions, scheduling, or to be added to the mailing
list, contact the organizers at:
probsematmathdottorontodotedu

PAST SEMINARS

Wednesday, June 30 at
10am 
Christian Sadel
(UC Irvine)
On the random phase conjecture for random Schroedinger operators
and
Joseph Najnudel (Univerisity of Zurich)
Permutations, virtual permutations, and a related flow
of operators

Monday,
June 28 at 4pm 
Mark Holmes (University of Auckland)
A combinatorial result with applications to random walks
Excited random walks (or random walks in cookie environments)
are random walk models in which the probability of departing
to the right from a site depends on the number of visits to
that site (equivalently how many cookies have been eaten at
that site) up to the present time. Work of Zerner, Basdevant
and Singh, and others, has demonstrated various different
types of behaviour possible with a bounded number of biased
cookies at each site
We'll discuss a deterministic combinatorial result that compares
two integer sequences l(n) and r(n) defined in terms of collections
L and R of arrows satisfying a natural monotonicity relation.
Applying this when the arrows are random, the sequences produce
nonmonotone couplings of selfinteracting random walks (such
as excited random walks in one dimension). This allows us
to extend known results about such models.

Tuesday,
July 21, 2009
2pm 
*SPECIAL SUMMER SEMINAR*
Tom Salisbury (York)
Random walks in degenerate random environments
At each vertex of the planar square lattice, independently
lay down pairs of arrows, heading either North and East (with
probability p), or South and West (with probability 1p).
Then do a simple random walk according to the available arrows.
This is one of a number of related RWRE models we'll consider.
An example of the results obtained is that for p close to
1/2 there is an infinite cluster of points that communicate
with each other, while for p close to 0 or 1 all such clusters
are finite. This is joint work with Mark Holmes (Auckland).

September 21, 2009 
Blint Virg(Toronto)
Random matrix eigenvalues and Gaussian analytic functions
There are two natural ways of producing repelling random
pointclouds on the complex plane: either by taking the eigenvalues
of some random matrix or by considering the zeros of a random
(Gaussian) analytic function. I will explain some old and
new connections between the two worlds.

September 28, 2009 
Domokos Szasz (Budapest University of Technology)
TBA

October 5, 2009 
Neal Madras (York University)
Entangled Clusters in Percolation
In bond percolation on the simple cubic lattice, each bond
is independently "open" with probability p. Suppose
we view each open bond as a solid but flexible bar, with all
bars that share an endpoint being joined at that point. Then
it is possible for two disjoint connected components to be
topologically entangled. Is it possible for all connected
components to be finite, and yet for an infinite number of
them to form a single entangled cluster? G. Grimmett and A.
Holroyd showed that this happens (almost
surely) for some values of p, but not when p is very close
to 0. They then asked whether the number of entangled clusters
(modulo translation) with exactly N edges is bounded exponentially
in N. We prove that the answer is yes. Among our corollaries
we obtain:
(1) an improved lower bound on the critical value for this
"entanglement percolation", and
(2) exponential decay of the tail probabilities for the size
of the entangled cluster containing the origin, when p is
small.This is joint work with Mahshid Atapour.

October 12, 2009 
** THANKSGIVING  NO
SEMINAR** 
October 19, 2009 
Domokos Szsz (Toronto and Budapest University of Technology)
Billiard Models and Energy Transfer
Recent progress in the theory of hyperbolic billiards has
enhanced the interest towards billiard models since they are
the most promising for the derivation of laws of statistical
physics from newtonian dynamics. Parallel to this progress,
energy transfer models (in particular, the study of diffusion
and of Fourier's law of heat conduction) have recently
come to the center of interest. In this talk I will present
two determinstic models with energy transfer and will treat
a stochastic version of one of them in detail.

October 26, 2009 
Paul Bourgade (Paris VI and Courant Institute)
Mesoscopic Fluctuations of Zeta Zeros
For large unitary matrices, the number of eigenvalues in
distinct shrinking intervals satisfies a central limit theorem,
whose covariance structure is related to some branching processes.
In this talk we present these random matrix results, following
works of Diaconis, Evans and Wieand, and show the strict analogue
for the zeros of Lfunctions.

November 2, 2009 
Gidi Amir (Toronto)
Liouville property, amenability, discrete fractals and automaton
groups
In this lecture we will discuss a phase transition in the
Liouville property for automaton groups.
We will start by discussing the Liouville property , which
says that the only bounded harmonic functions on a graph are
constant, and connect it to amenability of groups. We will
then describe automaton groups  which have played an important
role in providing (counter)examples to many problems in group
theory, (e.g. groups with intemediate growth) and their Schreier
graphs represent classical fractals such as the Sierpinski
gasket and Julia sets of finite polynomials such as the Basilica
set.
Our main result shows that there is a phase transition in
the Liouville property of automaton groups with respect to
a certain classification  the degree of activity growth 
that will be described. We conclude that all automaton groups
with up to linear activity are amenable. The proof has many
probablistic elements such as random walks on the Schreier
graphs and electrical networks.
This talk is based on joint work with O. Angel and B. Virag

November 9, 2009 
Jinho Baik (University of Michigan)
Maximal Crossings and Nestings of a Random Perfect Matching
A perfect matching on 2n is an n disjoint pairing of 2n numbers.
Given a matching, one can consider crossings and nestings.
For example, the number of perfect matchings with no crossings
equals the Catalan number, and so is the number of matchings
with no nestings. Recently, Chen, Deng, Du, Stanley and Yan
studied various properties of the maximal crossings and maximal
nestings of a matching. Especially they are shown to be symmetric
random variables if we consider uniformly random matchings.
It was also deduced, by making connections to various existing
works, that the maximal crossings and the maximal nestings
are asymptotically distributed as the GOE TracyWidom distribution
from random matrix theory. In this talk, we will show that
they are asymptotically independent. We use the determinantal
formula obtained by the above authors. Connections to an integrable
differential equation, the AblowitzLadik equation, will also
be discussed.
Joint work with Robert Jenkins.

November 16, 2009 
Alexey Kuznetsov
(York University)
WienerHopf factorization for Levy processes with meromorphic
characteristic exponent
WienerHopf factorization and related fluctuation identities
allow us to study various functionals of a Levy process, such
as extrema, first/last passage time, overshoot and undershoot,
etc. Due to many applications of Levy processes in Mathematical
Finance there is a lot of interest in studying processes for
which one can compute distributions of these functionals analytically.
Unfortunately the list of such examples is very short: it
contains processes with onesided jumps, subclass of stable
processes, processes with hyperexponential jumps. In this
talk we will discuss several recent results on WienerHopf
factorization for processes having meromorphic characteristic
exponent. This class is a natural generalization of processes
having rational transform; it shares many key properties related
to the analytical structure of WienerHopf factorization,
but at the same time it is a much larger class which allows
for more flexible modeling of jumps.

November 23, 2009 
Chuck Newman
(Courant Institute)
Ground states of the 2D EdwardsAnderson spin glass
It is an open problem to determine the number of infinitevolume
ground states in the EdwardsAnderson (nearest neighbor) spin
glass model on Z^d for d \geq 2 (with, say mean zero Gaussian
couplings). This is a limiting case of the problem of determining
the number of extremal Gibbs states at low temperature. In
both cases, there are competing conjectures for d \geq 3,
but no complete results even for d=2. I report on new results
which go some way toward proving that (with zero external
field, so that ground states come in pairs, related by a global
spin flip) there is only a
single ground state pair (GSP). Our result is weaker in two
ways: First, it applies not to the full plane Z^2, but to
a halfplane. Second, rather than showing that a.s. (with
respect to the quenched random coupling realization J) there
is a single GSP, we show that there is a natural joint distribution
on J and GSP's such that for a.e. J, the conditional distribution
on GSP's given J is supported on only a single GSP. The methods
used are a combination of percolationlike geometric arguments
with translation invariance (in one of the two coordinate
directions of the halfplane) and uses as a main tool the
``excitation metastate'' which is a probability measure on
GSP's and on how they change as one or more individual couplings
vary.
Joint work with LouisPierre Arguin, Michael Damron, and
Dan Stein.

November 30, 2009 
Emanuel Milman
(Toronto)
A stability result for the mixing time of Brownian motion
on a convex domain
It is classical that the $L_2$ mixing time of Brownian motion
on a nice domain in Euclidean space with reflectance on its
boundary, is inversely proportional to the spectral gap of
the Neumann Laplacian on that domain.
For convex domains, we provide a new characterization of
the spectral gap, by showing that Cheeger's isoperimetric
inequality, the spectral gap of the Neumann Laplacian, exponential
concentration of Lipschitz functions, and any arbitrarily
slow (but fixed) taildecay of these functions, are all quantitatively
equivalent (to within universal constants, independent of
the dimension). This extends previous results of Maz'ya, Cheeger,
GromovMilman, Buser and Ledoux. As an application, we conclude
a sharp quantitative stability result for the spectral gap
on convex domains under convex perturbations which preserve
volume (up to constants) and under maps which are ``onaverage''
Lipschitz. This also leads to the following characterization
of the square root of the $L_2$ mixing time of Brownian motion
on convex domains: it is equivalent (up to constants) to the
expectation (with respect to the uniform measure on the domain)
of the distance from the ``worst'' Borel set having measure
1/2. In addition, we easily recover (and extend) many previously
known lower bounds on the spectral gap of convex domains,
due to PayneWeinberger, LiYau, KannanLovaszSimonovits,
Bobkov and Sodin.
The proof involves estimates on the diffusion semigroup
following BakryLedoux and a result from Riemannian Geometry
on the concavity of the isoperimetric profile. Our results
extend to the more general setting of Riemannian manifolds
with density which satisfy the $CD(0,\infty)$ curvaturedimension
condition of BakryEmery, and to more general isoperimetric,
functional and concentration inequalities.

December 7, 2009 
INFORMAL SEMINAR
Laure Dumaz
Laura will speak about her master's thesis on the tails of the
TracyWidombeta distribution.

January 18, 2010 
Jeffrey Schenker
(Toronto)
Eigenvector Localization for Random Band Matrices with Power
Law Band Width
Random symmetric matrices with entries that vanish outside
a band around the diagonal, but otherwise have independent
identically distributed matrix elements, were introduced in
the physics literature as an effective model of a "localization/delocalization"
transition seen in disordered materials. In this talk, it
will be shown that such matrices satisfy a localization condition
which guarantees that eigenvectors have strong overlap with
only $W^\mu$ standard basis vectors where $W$ is the band
width and $\mu$ is an positive exponent. This statement is
vacuous if
$W^\mu> N$, the size of the matrix, but if $W^\m <<
N$ as $N$ tends to infinity, then a typical eigenvector is
essentially supported on a vanishing fraction of standard
basis vectors. For a Gaussian band ensemble, with matrix elements
given by i.i.d. centered Gaussians within a band of width
$W$, the estimate $\mu ~ ln 8$ holds. The role of this band
matrix model in physics and some open problems and conjectures
will also be discussed.

February 8, 2010 
Andreas Kyprianou
(University of Bath)
The Prolific Backbone of a Superdiffusion
We complete a result originally due to Evans and O'Connell in
1994 which equates the law of a supercritical, superdiffusion
with quadratic branching mechanism to that of the spatial tree
of a particle branching particle diffusion "dressed"
with immigrating subcritical superdiffusions. We give a direct
pathwise construction allowing for a fully general branching
mechanism. The key to our construction is the use of the DynkinKuznetsov
Nmeasures. This is based on joint work with Julien Berestycki
and Antoni Murillo.

February 22, 2010 
Brian Rider (University
of Colorado, Boulder)
Extremal laws in the real Ginibre matrix ensemble
The real Ginibre random matrix (the nxn matrix with all entries
iid standard normals) is perhaps the most natural nonHermitian
matrix ensemble one could ask for. Its spectrum however does
not enjoy the determinantal nature possessed by its complex
analog, making the determination of various spectral limit
laws more difficult. We show that, as expected from the complex
case, the spectral radius has a classical Gumbel distribution
limit. More interesting (maybe) we prove a limit law for the
largest real eigenvalue, described in terms of a Fredholm
determinant whose kernel (sadly) lacks some of the important
structure (in particular, integrability) that one is accustomed
to in random matrix theory.
Joint work with Christopher Sinclair.

March 1, 2010 
Alex Bloemendal
(Toronto)
The top eigenvalue of a spiked real sample covariance matrix:
A continuum operator limit approach
Simple trends in highdimensional data sets are modeled by
spiked real Gaussian sample covariance matrices. In the largesize
limit, the behaviour of the top eigenvalue exhibits a phase
transition as a function of the strength of then trend. Using
a stochastic operator limit approach, we show that the top
eigenvalue has an asymptotic distribution near the phase transition
and give several characterizations of the limit laws; one
of these involves only a linear boundary value problem. In
the wellstudied complex case, our PDE description reproduces
known explicit formulas and yields a simple new derivation
of the Painleve formula for the TracyWidom distribution.
This is joint work with Balint Virag.

Mar. 8, 2010 
Robert Masson (UBC)
Random walks on the two dimensional uniform spanning tree
We study random walks on the uniform spanning tree (UST)
on Z^2. We obtain estimates for the transition probabilities
of the random walk, the distance of the walk from its starting
point after n steps, and exit times of both Euclidean balls
and balls in the intrinsic graph metric. In particular, we
prove that the spectral dimension of the uniform spanning
tree on Z^2 is 16/13 almost surely.
In order to prove these results, we use the work of Barlow,
Jarai, Kumagai, Misumi and Slade on random walks on random
graphs which implies that it suffices to establish volume
and effective resistance bounds for the UST. Using Wilson's
algorithm, we show that this reduces to obtaining estimates
on the number of steps of looperased random walks (LERW)
in subsets of Z^2. If we let M_n be the number of steps of
a LERW from the origin to the circle of radius n, then Kenyon
showed that E[M_n] is logarithmically asymptotic to n^{5/4}.
In addition to this fact, we need
to show that with high probability, M_n is close to its mean.
In fact, we will obtain exponential moment bounds for M_n
which implies that the tails of M_n decay exponentially.
Joint work with Martin Barlow.

Mar. 15, 2010 
Ken Alexander (USC)
Disorderedpolymer depinning transitions: an overview
We consider a polymer with configuration modeled by the trajectory
of a Markov chain, interacting with a potential of form u+V_n
when it visits a particular state 0 at time n, with V_n representing
i.i.d. quenched disorder. There is a critical value of u above
which the polymer is pinned by the potential. The main question
of interest is, how does this depinning transition differ
from the one in the annealed model, where the interaction
is effectively homogeneous in n? The most important element
is the distribution of the return time to 0 for the underlying
Markov chain, and one would like to characterize those returntime
distributions which do and do not result in different critical
exponents, or critical points, for the quenched vs. annealed
model. This problem is much better understood than it was
5 years ago, though many questions remain open. We will give
an overview of some recent results.

**SPECIAL
FRIDAY SEMINAR: STEWART LIBRARY AT 3PM**
Mar. 19, 2010

Ori GurelGurevich
(Microsoft)
Nonconcentration of Return Times
Let T be the return time to the origin of a simple random
walk on a recurrent graph. We show that T is heavy tailed
and nonconcentrated. More precisely, we have
i) P(T>t) > c/sqrt(t)
ii) P(T=tT>=t) < C log(t)/t
Inequality i) is attained on Z, and we construct an example
demonstrating the sharpness of ii). We use this example to
answer negatively a question of Peres and Krishnapur about
recurrent graphs with the finite collision property (that
is, two independent SRW on them collide only finitely many
times, almost surely).
Joint work with Asaf Nachmias.

Mar. 22, 2010 
Mihai Stoiciu (Williams
College)
Random Matrices with Poisson Eigenvalue Statistics
We will give an overview of the recent results regarding
random matrices exhibiting local (microscopic) Poisson eigenvalue
statistics. It is known that the Poisson statistics holds
for certain classes of random Hermitian matrices, random unitary
matrices, random operators on rooted tree graphs and it is
conjectured that it also holds for random nonHermitian Anderson
models. We discuss in detail the case of the random CMV matrices,
which are the unitary analog of the onedimensional random
Schrodinger operator.

Mar. 29, 2010 
Miklos Abert (Courant)
Questions about Large Graphs of Random Degree
In the talk I will present a family of new, unsolved problems
and some partial solutions. The common theme is local (BenjaminiSchramm)
convergence of sequences of finite graphs of bounded degree.
We will use some notions of asymptotic group theory, ergodic
theory and topology.

Apr. 5, 2010 
Antonio Auffinger (Courant)
Directed Polymers in a Random Environment with Heavy Tails
We study the model of Directed Polymers in Random Environment
in 1+1 dimensions, where the environment is i.i.d. with a
site distribution having a polynomial tail with power \alpha,
where \alpha \in (0,2). After proper scaling of temperature
1/\beta, we show strong localization of the polymer to an
optimal region in the environment where energy and entropy
are best balanced. We prove that this region has a weak limit
under linear scaling and identify the limiting distribution
as an (\alpha, \beta) indexed family of measures on Lipschitz
curves lying inside the 45 degree rotated square with unit
diagonal. In particular, this shows order of n for the transversal
fluctuations of the polymer. If (and only if) \alpha is small
enough, we find that there exists a random critical temperature
above which the effect of the environment is not macroscopically
noticeable.
Joint work with Oren Louidor.

Apr. 12, 2010 
Jonathan Novak (Waterloo)
Wick formula for Haar unitary matrices
The Wick formula reduces the computation of Gaussian integrals
over a real or complex vector space to a combinatorial rule
plus knowledge of a scalar matrix (which probabilists call
the covariance matrix and physicists call the propagator).
Interesting applications of the Wick formula often deal with
Gaussian integrals over the space of Hermitian matrices, and
the associated combinatorics is that of maps on surfaces.
Now suppose that one wants to compute integrals over a compact
group of matrices, such as the unitary group. There is an
analogue of the Wick formula available in this setting  now
the "propagator" is the Gram matrix associated to
a certain projection. The resulting combinatorics is that
of symmetric polynomials evaluated at JucysMurphy elements,
and this connection allows one to obtain interesting asymptotics,
recursive structure, character expansion, and other features
of unitary matrix integrals.

***SPECIAL
DAY AND TIME***
Apr. 14, 2010
3:00 p.m.
Stewart Library 
Yuri Kifer
Nonconventional Limit Theorems
Abstract

Apr. 26, 2010

LouisPierre Arguin
(Courant)
Probabilistic Questions for Spin Glasses
For probabilists, a Gaussian spin glass is essentially a
Gaussian process on {1,+1}^N whose variance is N. An important
example is the SherringtonKirkpatrick model whose covariance
is a function of the Hamming distance on {1,+1}^N. In this
talk, I will give an overview of some of the conjectures coming
from physics about the behavior of the extrema in the limit
of large N, as well as the recent progress on these conjectures
in probability. I will focus especially on the ultrametricity
conjecture, which is a bold statement about the Gibbs measure
of these processes and their extremal statistics in general.

***SPECIAL
DAY AND TIME***
Apr. 14, 2010
3:00 p.m.
Stewart Library 
Yuri Kifer
Nonconventional Limit Theorems
Abstract 
Monday,
10 May 2010 1:00PM
Stewart Library 
The Doob htransform: theory and examples
Alex Bloemendal, University of Toronto
I will discuss what it means to condition a Markov process
on its exit state, and use the resulting theory to consider
various examples of conditioned random walk and Brownian motion.

Monday, 17 May 2010 at
1:00PM 
Doob's htransform: a few more examples
Alex Bloemendal, University of Toronto
Brownian motion conditioned never to exit a domain and the
ground state of the Dirichlet Laplacian; Weyl chambers, the
Vandermonde, and Dyson's Brownian motion; Polya's urn.

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