2011

Speaker and Talk Title

June 6, 2011,
4:10  5:00 PM, Stewart Library 
Fredrik Viklund (Columbia)
On Convergence Rates to SLE for Random Lattice Curves
The SchrammLoewner evolution (SLE) is a family of random
planar curves that appear as scaling limits of curves derived
from a range of discrete lattice models from statistical physics.
SLE is constructed by solving the Loewner differential equation
with a Brownian motion as the socalled Loewner driving function.
A first step towards proving convergence to SLE is to show
that the Loewner driving function for the discrete curve converges
to Brownian motion. In the talk we will discuss recent work
on how to estimate the convergence rate to the SLE curve,
given as input a convergence rate for the Loewner driving
function.

May 30, 2011,
4:10  5:00 PM, Stewart Library 
Mark Holmes (Auckland)
Random walks in degenerate random environments
In joint work with Tom Salisbury, we study random walks in
i.i.d. random environments in Z^d in dimensions 2 and higher.
In our environments, at any given site some steps may not
be available to the random walker (i.e. we don't assume ellipticity).
Among our main results are 01 laws for directional transience
(extending results already known under the assumption of ellipticity)
and a simple monotonicity result in for 2valued environments
(at each site the environment takes one of two values).

May 16, 2011,
4:10  5:00 PM, Stewart Library 
Mladen Savov (Oxford)
Exponential Functional of Lévy Processes
The law of the exponential functional of Lévy processes
plays a prominent role from both theoretical and applied perspectives.
We start this talk by describing some reasons motivating its
study and we review all known results concerning the distribution
of this random variable. We proceed by describing a new factorization
identity for the law of the exponential functional under very
mild con ditions on the underlying Lévy process.
As byproduct, we provide some interesting distributional
properties enjoyed by this random variable as well as some
new analytical expressions for its distribution (Joint work
with J.C. Pardo (CIMAT, Mexico) and P. Patie (Université
Libre de Bruxelles, Belgium)).

May 9, 2011,
4:10  5:00 PM, Stewart Library, 
Maurice Duits (Caltech)
An equilibrium problem for the two matrix model with a
quartic potential
In this talk, I will discuss the two matrix model in random
matrix theory and present some recent results on the asymptotic
behavior of
the eigenvalues statistics. In particular, a variational problem
will be introduced that characterizes the limiting eigenvalue
density for one of the matrices, in case one of the potentials
is quartic. I will also discuss the eigenvalue correlations
at the local scale and introduce a new universality class
near a multicritical point in the quartic/quadratic case.

May 2, 2011
Room 210
4pm 
Emily Redelmeier
(Queen's)
Fluctuations of large random matrices and secondorder freeness
Secondorder freeness extends free probabilistic approaches
to large random matrices from moments to fluctuations. Similarly
to the firstorder case, a definition of (complex) secondorder
freeness satisfied by independent ensembles of many important
matrix models (Ginibre, Wishart, unitarily invariant, Haardistributed
unitary) can be used as a rule for calculating fluctuations
of these matrices. However, the real analogues of these matrix
models do not generally satisfy this defintion. I will examine
the differences between the real and complex ensembles which
appear in some of the combinatorial tools applied to these
matrices, in particular the genus expansion, and present a
definition for real secondorder freeness satisfied by the
real matrix models.

April 26, 2011
Room 210
4pm 
Ilya Goldsheid (Queen
Mary, University of London)
Random transformations and related random walks in random
environment on a strip

*POSTPONED
TO APRIL 26*
April 18, 2011
Room 210
4pm 
Ilya Goldsheid (Queen
Mary, University of London)
Random transformations and related random walks in random
environment on a strip

April 11, 2011
Room 210
4pm 
Enza Orlandi (Universita' di Roma Tre)
Ginzburg Landau functional with external random field:
minimizers and interfaces
We add a random bulk term, modeling the interaction with
the impurities of the medium to a standard functional in the
gradient theory of phase transitions consisting of a gradient
term with a double well potential. We study the existence
and properties of minimizers.
The results strongly depend on dimensions and on the strength
of the random field. In d bigger or equal than 3 if the strength
of the random field is small enough there are a.s with respect
to the random field two minimizers and we compute the surface
tension of the interface. In dimensions d < 3 we show that
there exists only one minimizer and therefore no interfaces.
Joint work with Nicolas Dirr.

March 28, 2011
Room 210
4pm 
Gregorio Moreno Flores
(Fields & UWMadison)
Asymmetric directed polymers in a random environment
We study a model of very asymmetric directed polymers in
a random environment. We compute the free energy of the model
and the order of fluctuation of the partition function. As
in the very asymmetric last passage percolation, the key point
is an approximation by a Brownian percolation model, which
has strong connections with random matrices.

March 25, 2011
Room 210
11am 
Fraydoun Rezakhanlou
Stationary Processes, Symplectic Maps, and Arnold's Conjecture

March 25, 2011
Stewart Library
3:30pm 
Lincoln Chayes
(UCLA)
Equations of the McKeanVlasov type in finite volume
The McKeanVlasov system and its porous medium generalizations
will be described. These are nonlinear diffusion equations
with an additional nonlocal nonlinearity provided by convolution.
Recently popular in a variety of applications, these enjoy
an ancient heritage as a basis for understanding equilibrium
and near equilibrium fluids. The model is discussed in finite
volume where, on the basis of the physical considerations,
the correct scaling (for the model itself) is identified.
For dimension two and above and in large volume, various dynamical
anomalies are related to phase transitions; the phase structure
of the model is completely elucidated.

March 21, 2011
Stewart Library
4pm 
Milton Jara (IMPA)
Secondorder and local BoltzmannGibbs principle and applications
In the early '80s, Brox and Rost introduced the socalled
BoltzmannGibbs principle. As an application, they deduced
the time evolution of equilibrium fluctuations of the density
for interacting particle systems. In one dimension, we introduce
two generalizations of this principle, which we named secondorder
and local BoltzmannGibbs principle. As applications of these
generalizations, we prove that equilibrium fluctuations of
weakly asymmetric particle systems are given by energy solutions
of the KPZ equation, and we obtain novel functional limit
theorems for additive functionals of particle systems.
Joint with Patricia Gonalves (U. do MinhoPortugal).

March 16, 2011
Stewart Library
1pm 
Alan Hammond
Trapping of Biased Random Walk in Disordered Systems

March 14, 2011
Stewart Library
4pm 
Vadim Kaimanovich
Finite Approximation of Random Graphs
The theory of graphed equivalence relations provides a natural
point of view on random graphs. Namely, any invariant probability
measure on a graphed equivalence relation can be considered
as a probability measure on the space of rooted graphs. Benjamini
and Schramm showed that any weak limit of uniform measures
on finite graphs is an invariant measure on the space of rooted
graphs with respect to its natural equivalence relation. Recently
Elek proved that, conversely, any invariant measure on the
space of rooted graphs can be obtained in this way. We shall
show that this approximation property also holds for any invariant
measure on the space of rooted graphs such that a.e. graph
is Liouville (has no nonconstant harmonic functions).

March 7, 2011
Stewart Library
4pm 
Christophe Sabot
(Universite Lyon 1)
The environment viewed from the particle for random walks
in random Dirichlet environment
The environment viewed from the particle has been a powerful
tool in the investigation of random conductance models. For
nonreversible random walks in random environment the problem
of the equivalence of the static and dynamic points of view
is understood only in a few cases. The case of Dirichlet environment,
which corresponds to the case where the transition probabilities
at each site are iid Dirichlet random variables, is particularly
interesting since its annealed law corresponds to the law
of a reinforced random walk. In this talk, we will characterize,
for Dirichlet environments in dimension larger or equal to
3, the cases where the static and dynamic points of view are
equivalent. We can deduce from this a complete characterization
of the ballistic regimes in dimension larger or equal to 3.
The proof is based on crucial property of statistical invariance
by time reversal valid for the class of Dirichlet environments.

*SPECIAL SEMINAR
Mar. 1, 2011
N638 Ross Building, York University
10:30 a.m.

*SPECIAL SEMINAR
Lerna Pehlivan (Carleton University)
Top to random shuffles and number of fixed points
The number of fixed points is studied for random permutations
and for some shuffles such as riffle shuffles. We look at
the same problem for top to random shuffles. We provide the
formulas for the expected value and the variance of the number
of fixed points of a permutation obtained after a number of
top to random shuffles.

Feb. 28, 2011
Stewart Library
4pm 
Ivan Matic
Central Limit Theorems and Large Deviations for Variational
Problems
I will talk about variational problems related to the stochastic
HamiltonJacobi equations and its discrete analogues. Some
of these models have laws of large numbers and for them we
study the bounds on the variance and the large deviation events.

Feb. 21, 2011
Stewart Library
4pm 
Vladislav Vysotsky
Positivity of Integrated Random Walks
Consider the sequence of partial sums of a centered random
walk with finite variance. We study asymptotics of the probability
that the first $n$ terms of this sequence are positive, as
$n \to \infty$. The first result here is due to Ya. Sinai
(1992) who came to the problem considering solutions of the
Burgers equation with random initial data. The speaker's original
motivation emerged as these probabilities appeared in his
study of certain properties of sticky particle systems with
random initial positions. We present our results and discuss
the more general problem of finding small deviation probabilities
of integrated stochastic processes.

Feb. 11, 2011
Stewart Library
4pm 
Eric Cator (Delft)
Busemann functions and cube root fluctuations in the generalized
Hammersley process
I will explain the concept of Busemann functions in Last
Passage Percolation, and in particular for the generalized
Hammersley process. These Busemann functions turn out to be
very useful, for example to calculate asymptotic speeds of
multiple second class particles in an arbitrary rarefaction
intitial condition, but in this talk I will focus on how the
Busemann might be used to prove cube root fluctuations. For
the classical Hammersley process we have made these methods
rigorous.

Feb. 9, 2011
Stewart Library
3pm 
Francis Comets
Stochastic Billiards
A ball is moving at constant speed in straight line inside
a domain D of R^d, and bounces randomly upon hitting the boundary.
The sequence of impacts on th boundary is a natural random walk
on the boundary of D. For general bounded domains the walk is
ergodic. For the reflection law, the cosine density is of particular
interest, since the uniform measure on the boundary is invariant.
We consider also the case of unbounded domain, precisely the
case when D is an infinite "random tube". Under general
assumptions, the process is then diffusive in dimension d=3,4,...
The proof uses techniques from random media.
Joint work with Sergei Popov, Gunter Schutz, Marina Vachkovskaia.

Feb. 7, 2011
Stewart Library
4pm 
Sasha Sodin
Random Band Matrices
We shall discuss several conjectures regarding the spectral
properties of random band matrices, and some results that
can be proved using perturbation series.

***CANCELED***
Feb. 4, 2011
Stewart Library
4pm 
Eric Cator (Delft)
Busemann functions and cube root fluctuations in the generalized
Hammersley process
I will explain the concept of Busemann functions in Last
Passage Percolation, and in particular for the generalized
Hammersley process. These Busemann functions turn out to be
very useful, for example to calculate asymptotic speeds of
multiple second class particles in an arbitrary rarefaction
intitial condition, but in this talk I will focus on how the
Busemann might be used to prove cube root fluctuations. For
the classical Hammersley process we have made these methods
rigorous.

Jan. 24, 2011
Stewart Library
4:00 pm 
Alex Bloemendal
Finite rank perturbations of large random matrices
Finite (or fixed) rank perturbations of large random matrices
arise in a number of applications. The main phenomenon is
a phase transition in the largest eigenvalues as a function
of the strength of the perturbation. I will describe recent
and forthcoming work, joint with Balint Virag, in which we
introduce a new way to study such matrices. The main idea
is a reduction to a new banddiagonal form and the convergence
of this form to a continuum random Schroedinger operator on
the halfline. We describe the nearcritical fluctuations
in several ways, solving a wellknown open problem in the
real case. Another consequence is a new route to the Painleve
structure in the celebrated TracyWidom distributions.

Jan. 10, 2011
Stewart Library
4:00 pm 
Elena Kosygina
Crossing velocities for annealed random walks in random potentials
We consider random walks in an i.i.d. nonnegative potential
on the ddimensional integer lattice. The walks are conditioned
to hit a remote location and are studied under the annealed
path measure. When the potential is bounded away from zero,
it is very simple to show that the expected time needed by
the conditioned random walk to reach a remote location, call
it y, grows at most linearly in y. The question becomes
much harder if the potential is allowed to be zero with positive
probability. We prove that even in this situation the expected
time to reach y increases only linearly in y. In dimension
one we can show the existence of the asymptotic speed as y
goes to infinity.
The motivation for this question comes from an attempt to
compare Lyapunov exponents and, thus, quenched and annealed
large deviations rate functions for random walks in small
potentials, that are not bounded away from zero.
This is a joint work with Thomas Mountford (EPFL, Lausanne).

PLEASE
NOTE DATE AND TIME
Jan. 5, 2011
Stewart Library
4:00pm

Martin Zerner
Interpolation Percolation
We consider a twodimensional infinitesimal continuum percolation
model with columnar dependence. It is related to oriented
percolation, firstpassage percolation, Lipschitz percolation,
Poisson matchings and coverings of the circle by random arcs.
Several open questions are posed.

Dec.6, 2010
Stewart Library
4:00pm

CANCELLED
Fredrik Viklund (Columbia University)
TBA

PLEASE NOTE DATE AND TIME
Nov. 26, 2010
Stewart Library
2:40pm

Sunil Chhita (Brown University)
Particle Systems Arising from an AntiFerromagnetic Ising
Model
We present a low temperature anisotropic antiferromagnetic
2D Ising model through the guise of a certain dimer model.
This model also has a bijection with a onedimensional particle
system equipped with creation and annihilation. We can find
the exact phase diagram, which determines two significant
values (the independent and critical value). We also highlight
some of the behavior of the model in the scaling window at
criticality and at independence.

PLEASE
NOTE DATE AND TIME
Nov. 26, 2010
Stewart Library
4:10pm 
Leonid Koralov (University
of Maryland)
Nonlinear Stochastic Perturbations of Dynamical Systems
We will describe the asymptotic behavior of solutions to
quasilinear parabolic equations with a small parameter at
the second order term and the long time behavior of corresponding
diffusion processes. In particular, we discuss the exit problem
and metastability for the processes corresponding to quasilinear
initialboundary value problems.

Nov. 15, 2010
Stewart Library
4pm 
Gérard Letac
(Universit Paul Sabatier)
Meixner Random Matrices
M is a Meixner probability on R if for X and Y independent
with distribution M and if S=X+Y then the conditional expectation
of X*2 knowing S is a quadratic polynomial in S. There are
6 types of them: Bernoulli, Poisson, negative binomial, Gaussian,
gamma and hyperbolic. In this lecture we consider the same
problem when M is a probability on the (n,n) symmetric matrices
or more generally on Hermitian complex or quaternionic 
invariant by rotation. We find back the six types again. For
instance the Bernoulli type is obtained as the mixing of the
distributions Mk for k=0,1,...,n where Mk is the law of the
orthogonal projection on a uniformly distributed random subspace
of dimension k. The Laplace transforms of these Meixner distributions
are characterized by a linear system of PDE with a finite
dimensional set of solutions.
This is joint work with W. Bryc.

Nov. 8, 2010
Stewart Library
4pm 
Pierre Nolin (Courant Institute)
Connection probabilities and RSWtype bounds for the twodimensional
FK Ising model
For twodimensional independent percolation, RussoSeymourWelsh
(RSW) bounds on crossing probabilities are an important apriori
indication of scale invariance, and they turned out to be
a key tool to describe the phase transition: what happens
at and near criticality.
In this talk, we prove RSWtype uniform bounds on crossing
probabilities for the FK Ising model at criticality, independent
of the boundary conditions. A central tool in our proof is
Smirnov's fermionic observable for the FK Ising model, that
makes some harmonicity appear on the discrete level, providing
precise estimates on boundary connection probabilities.
We also prove several related results  including some new
ones  among which the fact that there is no magnetization
at criticality, tightness properties for the interfaces, and
the value of the halfplane onearm exponent.
This is joint work with H. DuminilCopin and C. Hongler.

Nov. 1, 2010
Stewart Library
4pm 
Dmitry Jakobson
(McGill University)
Gauss Curvature of Random Metrics
We study Gauss curvature for random Riemannian metrics on
a compact surface, lying in a fixed conformal class ``close''
to a fixed (background) metric.
We explain how to estimate the probability that Gauss curvature
will change sign after a random conformal perturbation of
a metric; discuss some extremal problems for that probability,
and their relation to other extremal problems in spectral
geometry.
Generalizations to higher dimensions will be discussed in
my talk at the Workshop on Geometric Probability and Optimal
Transportation on Wednesday, November 3, in Room 230, 2:10
 3:00.
This is joint work with Y. Canzani and I. Wigman

Oct. 25, 2010
Stewart Library
4pm 
Leonid Pastur (National Academy of Sciences of Ukraine/Fields)
Limiting Fluctuation Laws for Spectral Statistics of Random
Matrices

Oct. 18, 2010
Stewart Library
4pm 
Misha Sodin (Tel Aviv University)
Random complex zeros: fluctuations and correlations
In the talk, I plan to discuss our recent results with Fedja
Nazarov (arXiv:1005.4113, arXiv:1003.4251): close to optimal
conditions on a testfunction that yield asymptotic normality
of the corresponding linear statistics of random complex zeroes;
universal local bounds for kpoint functions of zeroes, and
their strong clustering.

Sept. 27, 2010
Stewart Library
4pm 
Daniel Remenik (University of Toronto)
BrunetDerrida particle systems, free boundary problems,
and WienerHopf equations
We consider a branchingselection system in R with $N$ particles
which give birth independently at rate 1 and where after each
birth the leftmost particle is erased, keeping the number
of particles constant. We show that, as $N\to\infty$, the
empirical measure process associated to the system converges
in distribution to a deterministic measurevalued process
whose densities solve a free boundary integrodifferential
equation. We also show that this equation has a unique traveling
wave solution traveling at speed $c$ or no such solution depending
on whether $c\geq a$ or $c<a$, where $a$ is the asymptotic
speed of the branching random walk obtained by ignoring the
removal of the leftmost particles in our process. The traveling
wave solutions correspond to solutions of WienerHopf equations.
Joint work with Rick Durrett.

Sept. 20, 2010
Stewart Library
4pm 
Alan Hammond (Oxford University)
Biased motion in disorder: persistent discreteness, rational
resonance, and stable limits
A biased random walker in open space will move at positive
velocity. If the medium is disordered, however, the motion
may be slowed to vanishing velocity by the walker encountering
large connected structures in the disorder that acts as traps.
A natural model for these effects is a walker on an infinite
GaltonWatson tree with leaves, with a constant bias away
from the root. Here, the finite trees hanging off the backbone
act as traps. The progress of the walker is determined on
all timescales by a discrete inhomogeneity, in which trap
sojourn times tend to cluster around powers of the bias parameter.
This prevents the existence of a scaling limit. I will introduce
an alternative model, in which biases on edges of the tree
are randomized with a nonlattice distribution, so that a
stable limiting law results. These two effects, of persistent
discrete inhomogeneity in a constant bias model, and stable
limiting laws in the randomly biased case, may have counterparts
in more physical models in Euclidean space, where the persistent
discreteness may arise as a rational resonance in the bias
slope.
Coauthors: Alex Fribergh (Z^d and tree models), Gerard Ben
Arous and Nina Gantert (tree models).

Sept. 13, 2010
Stewart Library
4pm 
Tom Alberts (University of Toronto)
Intermediate Disorder for Directed Polymers in Dimension
1+1, and the Continuum Random Polymer
The 1+1 dimensional directed polymer model is a Gibbs measure
on simple random walk paths of a prescribed length. The weights
for the measure are determined by a random environment occupying
spacetime lattice sites, and the measure favors paths to
which the environment gives high energy. For each inverse
temperature $\beta$ the polymer is said to be in the weak
disorder regime if the environment has little effect on it,
and the strong disorder regime otherwise. In dimension 1+1
it turns out that all positive $\beta$ are in the strong disorder
regime. I will introduce a new regime called intermediate
disorder, which is accessed by scaling the inverse temperature
to zero with the length $n$ of the polymer. The precise scaling
is $\beta n^{1/4}$. The most interesting result is that under
this scaling the polymer has diffusive fluctuations, but the
fluctuations themselves are not Gaussian. Instead they are
still coupled to the random environment, and their distribution
is intimately related to the TracyWidom distribution for
the largest eigenvalue of a random matrix from the GUE. More
recent work also shows that we can take a scaling limit of
the entire intermediate disorder regime to construct a continuous
random path under the effect of a continuum random environment.
We call the scaling limit the continuum random polymer. I
will discuss a few properties of the continuum random polymer
and its intimate connection to the stochastic heat equation
in one dimension.
Joint work with Kostya Khanin and Jeremy Quastel.

July 30, 2010
11:10 am
Stewart Library 
Dong Wang (University
of Michigan)
Hermitian matrix model with spiked external source
The Hermitian matrix model is usually analyzed by a RiemannHilbert
problem of size higher than 2. If the external source is spiked,
i.e., only finitely many eigenvalues of the external source
matrix are nonzero, we show a new approach to solve the problem.
First we solve the rank 1 case by steepestdescent method,
and then by a determinantal formula we derive the result in
the higher rank case from that in the rank 1 case. We show
the asymptotic behavior of the largest eigenvalue. Joint work
with Jinho Baik.

July 19, 2010
4:10 pm
Stewart Library 
Arnab Sen (UC Berkeley)
Coalescing systems of nonBrownian particles
A wellknown result of Arratia shows that one can make rigorous
the notion of starting an independent Brownian motion at every
point of an arbitrary closed subset of the real line and then
building a setvalued process by requiring particles to coalesce
when they collide. Arratia noted that the value of this process
will be almost surely a locally finite set at all positive
times, and a finite set almost surely if the starting set
is compact. We investigate whether such instantaneous coalescence
still occurs for coalescing systems of particles where either
the state space of the individual particles is not locally
homeomorphic to an interval or the sample paths of the individual
particles are discontinuous. We show that Arratia's conclusion
is valid for Brownian motions on the Sierpinski gasket and
for stable processes on the real line with stable index greater
than one.
Joint work with Steve Evans and Ben Morris.

July 5, 2010
4:10 pm
Stewart Library 
Ivan Corwin (CourantNYU)
Fluctuations for the KPZ Universality Class
We consider the weakly asymmetric limit of simple exclusion
with drift to the left, starting with step Bernoulli initial
data so that macroscopically one has a rarefaction fan. We
study the fluctuations of the associated height function process
observed along slopes in the fan, which are given by the HopfCole
solution of the KardarParisiZhang equation, with appropriate
initial data. Slopes strictly inside the fan correspond with
Dirac delta function initial data, while at the edge of the
rarefaction fan, the initial data is one sided Brownian. We
provide exact formulas for the one point distributions of
these KPZ fluctuations which, as time goes to infinity, recover
the expected TracyWidom type limit.
