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PAST SEMINARS 2011-2012
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Friday, May 4
3:10 pm
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Lukasz Grabowski (Imperial College, London)
Decidability aspects of computing spectral measures
Given a finitely generated group G we can fix a generating
set g_1, g_2, ... g_n and consider T to be a random walk (or
more general convolution operator) on the Cayley graph of
G wrt the generators g_1, ..., g_n. In the talk we will investigate
computational problems related to computing the spectral measure
of T: in particular, is there an algorithm which answers the
question ``is the kernel of T non-trivial?" I will give
many examples of groups where there is such an algorithm and
sketch a proof why there is no such algorithm for the group
$H^4$, where H is the lamplighter group $Z_2 \wr Z$.
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May 7
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Alejandro Ramirez (Pontificia Universidad catolica
de Chile)
Criteria for ballistic behavior of random walks in random
environment
It is conjectured that for a random walk on the hipercubic
lattice Z^d in a uniformly elliptic i.i.d. random environment,
when the dimension d ? 2, transience in a given direction
implies ballisticity in the same direction. In this talk,
I will review some recent progress on this question made in
collaboration with Noam Berger, David Campos and Alexander
Drewitz. In particular, I will introduce a set of polynomial
ballisticity criteria and several renormalization methods.
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April 16
Fields Institute
Room 230 |
Janosch Ortmann (University of Warwick)
Product-form invariant measures for Brownian motion with
drift satisfying a skew-symmetry type condition
Motivated by recent developments on positive-temperature
polymer models we propose a generalisation of reflected Brownian
motion (RBM) in a polyhedral domain. Our process is obtained
by replacing the singular drift on the boundary by a continuous
one which depends, via a potential U, on the position of the
process relative to the domain. We show that our generalised
process has an invariant measure in product form if we have
a certain skew-symmetry condition that is independent of the
choice of potential. Applications include a exponential analogue
of the Brownian TASEP, examples motivated by queueing theory,
Brownian motion with rank-dependent drift and a process with
close connections to the \delta-Bose gas.
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April 2
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Benedek Valko
(UW-Madison)
Scaling exponents of lattice gases
We consider a a family of lattice gas models (speed change
models) where the particles move randomly according to a local
rule with the extra condition that there is at most one particle
at any site. We study the equilibrium fluctuations at a given
particle density, in particular we are interested in the scaling
exponents of certain physical quantities. These exponents
are predicted to be governed by the local behavior of the
macroscopic flux function at the equilibrium density. We will
give upper and lower bounds on these exponents, in particular
we will confirm the superdiffusive behavior of the models
in all the cases where this was predicted by physical arguments.
Joint with Jeremy Quastel (Toronto).
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March 14
3:10 p.m
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Perla Sousi (University
of Cambridge)
The effect of variable drift on Brownian motion and the Wiener
sausage
The Wiener sausage at time t is the algebraic sum of a Brownian
path on [0,t] and a ball.Does the expected volume of the Wiener
sausage increase when we add drift? How do you compare the
expected volume of the usual Wiener sausage to one defined
as the algebraic sum of the Brownian path and a square (in
2D) or a cube (in higher dimensions)? We will answer these
questions using their relation to the detection problem for
Poisson Brownian motions, and rearrangement inequalities on
the sphere. (Talk based on joint works with Yuval Peres)
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March 16
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CANCELLED:
Van Vu (Yale University)
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March 5
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James Nolen
(Duke University)
Normal approximation for a random elliptic PDE
I will talk about solutions to an elliptic PDE with conductivity
coefficient that varies randomly with respect to the spatial
variable. It has been known for some time that homogenization
may occur when the coefficients are scaled suitably. Less is
known about fluctuations of the solution around its mean behavior.
For example, if an electric potential is imposed at the boundary,
some current will flow through the material. What is the net
current? For a finite random sample of the material, this quantity
is random. In the limit of large sample size it converges to
a deterministic constant. I will describe a central limit theorem:
the probability law of the energy dissipation rate is very close
to that of a normal random variable having the same mean and
variance. I'll give an error estimate for this approximation
in total variation. |
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Feb. 27
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Yuri Bakhtin
(Georgia Tech)
Randomly forced Burgers equation in noncompact setting
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. In this talk, I will discuss the quasi-compact
case where the random forcing decays to zero at infinity and
the completely noncompact case of forcing that is stationary
in space-time. The main result is the description of the ergodic
components for the dynamics and One Force One Solution principle
on each of the components. Joint work with Eric Cator and Kostya
Khanin.
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Feb. 15
5:00 p.m
*Please not non-standard date and time
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Charles Bordenave
(Université de Toulouse)
Localization and delocalization of eigenvectors for heavy-tailed
random matrices
This is a joint work with Alice Guionnet. Consider an n
x n Hermitian random matrix with, above the diagonal, independent
entries with alpha-stable symmetric distribution and 0 <
alpha < 2. The spectrum of this random matrix differs significantly
from the spectrum of Wigner matrices with finite variance. It
can be seen as an instance of a sparse random matrix : only
O(1) entries in each row have a significant impact on the behavior
of the matrix. In this talk, we will give bounds on the rate
of convergence of the empirical spectral distribution of this
random matrix as n goes to infi nity. When 1 < alpha <
2 and p large enough, we will see that the Lp-norm of the eigenvectors
normalized to have unit L2-norm goes to 0. On the contrary,
when 0 < alpha < 2/3, we will see that these eigenvectors
are localized. These localization/delocalization results only
partially recover some predictions due to Bouchaud and Cizeau
in 1994. |
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Feb. 6
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Daniel Remenik (University
of Toronto)
Variational formulas for the Airy2 process
The Airy2 process arises as the scaling limit of the fluctuations
in a variety of one dimensional random growth models with curved
initial conditions and point-to-point directed random polymers,
all in the KPZ universality class. In this talk I will explain
how to get a formula for the continuum statistics of the Airy2
process or, more precisely, for the probability that the process
lies below a given function on some finite interval. Then I
will explain how this formula can be used to confirm some predictions
which relate a variational problem for the Airy2 process to
certain asymptotic distributions in the KPZ universality class
associated with other initial conditions. I will also explain
how the formula can be used to compute the asymptotic endpoint
distribution for point-to-line polymers. This is joint work
with Ivan Corwin, Gregorio Moreno Flores and Jeremy Quastel.
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Jan. 30
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Karl Liechty (University
of Michigan)
Nonintersecting processes on the half-line and discrete orthogonal
polynomials
We will consider a system of n Brownian motions on the time
interval [0,1] taking values the nonnegative real numbers such
that all of them begin at zero at time t=0, and return to zero
at time t=1, and such that they never intersect for t \in (0,1).
The condition that the Brownian motions are never negative can
be thought of as a "wall" at zero. In the proper scaling,
the distribution of the top curve should converge to the Airy
process, and thus the distribution of the maximum value of the
top curve should converge to the Tracy-Widom distribution for
the Gaussian orthogonal ensemble. I will formulate the distribution
of the maximum value of the top curve in terms of a system of
discrete orthogonal polynomials which then can be evaluated
in an appropriate double scaling limit. This double scaling
limit corresponds to the system of orthogonal polynomials approaching
saturation, meaning that their zeroes become as tightly packed
as possible. If time permits, I will also discuss a discrete
version of the same problem.
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Jan. 23
3:10 p.m
**Please note non-standard time
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Raoul Normand (Paris
VI, Toronto)
Self-organized criticality in a discrete model of coagulation
The goal of this talk is to present a random model of coagulation,
meant to be a discrete model for Smoluchowski's equation. Loosely,
one starts with a large number of particles, each with a certain
number of arms, used for the coagulations. Then we pair, at
each step, two arms chosen uniformly at random, but only in
"small" clusters. We want to study the shape of the
clusters in this model, and we will explain how it exhibits
a phenomenon of self-organized criticality. This is a joint
work with Mathieu Merle (LPMA, Univ. Paris VII). |
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Jan.16
3:10 p.m
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Almut Burchard (University
of Toronto)
Random sequences of symmetrizations
In geometric optimization problems, it can be useful to approximate
the symmetric decreasing rearrangement by a sequence of simpler
rearrangements, such as polarization
(two-point rearrangement), Steiner symmetrization, or the Schwarz
rounding process. In the literature, considerable effort goes
into the construction of special sequences
of rearrangements that yield full rotational symmetry in the
limit. Here, I will discuss conditions under which random sequences
of rearrangements converge almost surely to the
symmetric decreasing rearrangement and give bounds on the rate
of convergence. I will also show examples how convergence can
fail. The talk is based on results from Marc Fortier's 2010
M.S. thesis and more recent results, including work in progress
with Gabriele Bianchi, Paolo Gronchi, and Ajosa Volcic.
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Jan. 9
3:10 p.m
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Codina Cotar (Fields Institute)
Uniqueness of random gradient states
We consider two versions of random gradient models. In
model A) the interface feels a bulk term of random fields
while in model B) the disorder enters though the potential
acting on the gradients itself. It is well known that without
disorder there are no Gibbs measures in
infinite volume in dimension d = 2, while there are gradient
Gibbs measures describing an infinite-volume distribution
for the increments of the field, as was shown by Funaki and
Spohn. Van Enter and Kuelske proved that adding a disorder
term as in model A) prohibits the existence of such gradient
Gibbs measures for general interaction potentials in d = 2.
Cotar and Kuelske proved the existence of shift-covariant
gradient Gibbs measures for model A) when d\ge 3 and the expectation
with respect to the disorder is zero, and for model B) when
d\ge 2. In the current work, we prove uniqueness of shift-covariance
gradient Gibbs measures with given tilt under the above assumptions.
(this is joint work with Christof Kuelske).
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Dec. 16
3:10 p.m
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Christian Sadel (UC Irvine)
A multi-channel 1D random Dirac operator with purely
absolutely continuous spectrum
For Anderson-type random operators one expects the existence
of absolutely continuous spectrum for low disorder in 3 and
higher dimensions. In contrast, one-dimensional structures
usually have pure point spectrum because the Lyapunov exponents
are positive. However, certain symmetries in the model can
lead to zero Lyapunov exponents and absolutely continuous
spectrum. We will give such an example in the form of a multi-channel
Dirac operator with random matrix potential with time-reversal
symmetry.
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Dec. 9
3:10 p.m
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Robert Young (University of Toronto)
Pants decompositions of random surfaces
Random graphs often have remarkable geometric and combinatorial
properties, but what about random high-genus surfaces? In
this talk, I will describe a construction of a random high-genus
surface and prove one of its unusual geometric properties.
No previous knowledge of the geometry of surfaces is necessary.
This is joint work with Larry Guth and Hugo Parlier.
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Dec. 2
3:10 p.m
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Arnab Sen, Cambridge University
Reverse hypercontractivity and its applications
An operator T is said to satisfy a hypercontractive inequality
if there exist q > p > 1 (depending on T) such that |Tf|_q
\leq \|f\|_p for all functions f. Hypercontractive inequalities
are extremely well
known and play a fundamental role in discrete harmonic analysis
as well as other areas of mathematics. An operator T is said
to satisfy a reverse hypercontractive inequality if there exist
q < p < 1 (q and p can be negative) such that |Tf|_q \geq
|f|_p for all strictly positive functions f. Reverse hypercontractive
inequalities started to emerge in recent years as a useful tool
for providing solutions to a number of problems. In this talk
I will discuss some new results relating reverse hypercontractive
inequalities to hypercontractive, Log-Sobolev and Poincare inequalities
in the setting of finite reversible Markov chains. I will also
describe some of the old and new applications of the reverse
hypercontractive inequalities.
This is joint work with Elchanan Mossel and Krzysztof Oleszkiewicz.
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2:00 p.m Probability Study Group BA 6180
Raoul Normand, Paris 6, Toronto
A model of migration under constraints
The goal of this talk is to present a simple model of
population with migration. The tools used are classical, namely
Galton-Watson trees, random walks and Brownian motion, so
this talk is accessible to graduate students.
In our model, migrations are constrained, in that people
migrate when the resources on the "island" where
they live are exhausted. From the genealogy of an individual
(a Galton-Watson tree), we can construct the "tree of
isles", describing the genealogy of the migrations. A
first step is to describe this tree. The second step is to
study limits. The relevant parameters are encoded in a random
walk, and thus the limiting quantities are related to the
Brownian motion.
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Nov. 25
3:10 p.m.
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Carl Mueller, Rochester
Uniqueness and nonuniqueness for some stochastic PDE
Good uniqueness criteria for stochastic differential equations
have been known for a long time, at least in the one dimensional
case. The situation is very different for stochastic PDE (SPDE),
and uniqueness criteria for parabolic SPDE have only appeared
recently. We will discuss some of this progress, focusing on
equations related to the superprocess, which is a limit of branching
Brownian motions.
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Nov. 18
2:10 pm
Room 6180, Bahen
*Please note non-standard time and location
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Affliated Actvity
Probability Study Gruop
Codina Cotar
will give the first of three lectures on Gradient Models
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Nov. 18
3:10 pm
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Joseph Najnudel
(Zurich)
A unitary extension of virtual permutations
The virtual permutations, introduced by Kerov, Olshanski
and Vershik, are sequences of permutations of increasing order,
whose cycle structure satisfies some compatibility properties.
If these permutations are chosen uniformly and considered
as random matrices, then one can prove an almost sure convergence
of their renormalized eigenangles. In a joint paper with P.
Bourgade and A. Nikeghbali, we extend this result to sequences
of matrices following Haar measure on the unitary group.
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Nov. 16
3:10 pm
*Please note non-standard time
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Jeremy Quastel (University of Toronto)
Variational problems for Airy processes
We show how to compute the joint density of the max and argmax
of the Airy_2 process minus a parabola. The argmax is a universal
distribution governing the endpoint of directed random polymers
in 1+1 dimensions.
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Nov. 11
2:30 pm
*Please note non-standard time
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Thomas Bloom (University
of Toronto)
Large deviations for random matrices via potential theory
Ben Arous and A.Guionnet gave the first Large Deviation result
for the Gaussian Unitary Ensemble.(drawing on work of Voiculescu).
Their method was subsequently extended to general Unitary
and other ensembles.I will outline a new proof, using potential
theory, of those large deviation results. I will also discuss
a large deviation result for a multivariable generalization
of Unitary ensembles. (This uses recent developments in pluripotential
theory due to R.Berman and S.Boucksom.) No prior knowledge
of potential or pluripotential theory will be assumed.
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Nov. 4
3:10 pm
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Robert McCann (Toronto)
Imagine some commodity being produced at various locations
and consumed at others. Given the cost per unit mass transported,
the optimal transportation problem is to pair consumers with
producers so as to minimize total transportation costs. Despite
much study, surprisingly little is understood about this problem
when the producers and consumers are continuously distributed
over smooth manifolds, and optimality is measured against
a cost function encoding some geometry of the product space.
This talk will include an introduction to the optimal transportation,
its relation to Birkhoff's problem of characterizing of extremality
among doubly stochastic measures, and some recent progress
linking the two. In particular, we expose the topology of
the cross-difference, which explains why extremal measures
concentrate on thin sets, and which underlies a criterion
for uniqueness of solutions subsuming all previous criteria,
and which is among the very first to apply to smooth costs
on compact manifolds, yet remains limited to topological spheres.
***Probability Study Group 2:00-3:00 pm
BA6180.
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Oct. 28
3:10 pm
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Ben Rifkind (Toronto)
A Stochastic Version of Multiplicative Cascades
I will discuss a 1 dimensional version of a random geometry
know as Multiplicative Cascades. In 2008, Benjamini and Schramm
proved a KPZ relation which describes this random geometry relative
to the regular Euclidean one. In joint work with Tom Alberts,
we study the evolution of this random geometry in time and extract
an ODE for the KPZ formula.
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Oct. 21
3:10 pm
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Balint Virag (Toronto)
Finite graphs and Kesten's theorem
Kesten showed that any transitive infinite d-regular graph
whose spectral radius is minimal is a tree. In this talk,
we present a version of this theorem for general finite d-regular
graphs: if the spectral radius is close to minimal, then a
large neighborhood around most points is a tree.
This is a joint work with M. Abert and Y. Glasner.
BA6290B (small meeting room across from the math library)
from 2-3 on Friday.
Let I_N denote the size of the largest independent set of
the Erdos-Renyi random graph
G(N,cN) consisting of N vertices and chosen uniformly at random
from the set of all graphs on N vertices with cN edges. It
was conjectured that I_N/N converges to a limit almost surely,
and this was recently proved by Bayati, Gamarnik and Tetali
using a clever combinatorial interpolation argument. I will
present their argument, which was also used to prove almost
sure scaling limits for other combinatorial problems on random
graphs.
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***Probability Study Group
2:00 - 3:00 p.m.
*Please note this event is at the Bahen Centre
BA6290B (small meeting room across from the math library)
Let I_N denote the size of the largest independent set of
the Erdos-Renyi random graph G(N,cN) consisting of N vertices
and chosen uniformly at random from the set of all graphs
on N vertices with cN edges. It was conjectured that I_N/N
converges to a limit almost surely, and this was recently
proved by Bayati, Gamarnik and Tetali using a clever combinatorial
interpolation argument. I will present their argument, which
was also used to prove almost sure scaling limits for other
combinatorial problems on random graphs.
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Oct. 14
3:10 pm
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Narn-Rueih Shieh (Taipei)
Scaling Limits for some PDEs with Random Initial Data
Let X(x,t) (for x in R^n and t in R+) be the spatial-temporal
random field arising from a certain parabolic PDE with initial
data given by a subordinated Gaussian field. We discuss the
scaling limit of such a space-time random field. The space-time
correlation structure of the solution field and the Hermite
expansion associated with the initial data play the essential
roles in our study.
Similar results hold for X an R^2-valued spatial-temporal
random field arising from a certain two-equation system, under
very weak coupling. Scaling limits for time-fractional and
spatial-fractional systems are also reported. There are some
novel features in the fractional case.
This talk is based on joint work with G.-R. Liu (a PhD student
in Taiwan)
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Sept. 12
3:10 pm
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Balazs Szegedy
Graph limits and corresponding spectral theory
The foundation of the so-called graph limit theory is a certain
compactifiction of the set of finite graphs which captures
both local (weak) and global (Szemeredi partition) convergence.
The purpuse of this talk is to extend the spectral theory
of finite graphs to the graph limit space. Along these lines
we prove a spectral version of Szemeredi's regularity lemma.
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