
Toronto Probability Seminar 200708
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

2008

Speaker and Talk Title

June 16
4:105
Fields Library 
Eckhard Schlemm, FU Berlin (visiting U
of T)
will present a talk about his masters thesis (Diplomarbeit)
on Firstpassage percolation on widhtwo stretches

Tuesday,
April 8, 2008
4:30 p.m.
215 Huron,
Room 1018
*Note Unusual Time and Place*

Mate Matolcsi (Renyi Institute of Mathematics, Hungary)
The real polarization problem
We study a conjecture of Benitez, Sarantopoulos and Tonge
concerning a lower bound on the norm of products of real linear
functioanls. The conjecture is that the lowerbound is attained
if and only if the vectors corresponding to the functionals
are orthogonal. There are several approaches to the problem,
analytic (Revesz, Pappas, 2004), geometric (Matolcsi, 2005),
and probabilistic (Frenkel, 2007), yielding partial results.
The probabilistic approach of Fernkel, 2007, deduces a lower
bound from the following theorem: If X1, ... , Xn are jointly
Gaussian random variables with zero expectation, then E(X1^2
... Xn^2) >= EX1^2 ... EXn^2. Equality holds if and only
if they are independent or at least one of them is almost
surely zero. A similar result for higher moments would imply
the conjecture.

Monday,
March 24, 2008 4:00 pm,
Stewart Libary Fields 
Lincoln Chayes, UCLA
On the absence of ferromagnetizm in typical 2D ferromagnets. 
Monday,
March 17, 2008 10:10 am,
215 Huron St. 
B. Valko and B. Virag, University
of Toronto
The Brownian Carousel
In the fourth and final part of this epic trilogy we explain
some details of the proof of that connects random matrices
to hyperbolic Brownian motion. 
Monday,
March 10, 2008 4:00 pm,
Stewart Libary Fields 
B. Valko and B. Virag, University of Toronto
The Brownian Carousel, part 2b.
The eigenvalues of a random Hermitian matrix form a random
set of points on the real line. As the matrix size converges
to infinity, the eigenvalues, after appropriate scaling, converge
to a point process.
The possible limit processes, called Sinebeta processes,
are fundamental objects of probability theory. They are famous
for their conjectured relationship to the Riemann zeta zeros,
Dirichlet eigenvalues of Euclidean domains, random Young tableaux,
and noncolliding walks. This series of informal talks is
about a new description of these processes in terms of Brownian
motion in the hyperbolic plane, called the Brownian carousel.
We plan to have three lectures:
1. Introduction to random matrix eigenvalues, definition and
basic properties of the Brownian Carousel
2. Computing with the Brownian carousel; continuity, phase
transitions, Dyson's predictions
3. Convergence of finite random matrix eigenvalues to the
Brownian carousel

Monday,
March 3, 2008 4:00 pm,
Room TBA 
B. Valko and B. Virag, University
of Toronto
The Brownian Carousel, part 2 
Monday,
Feb. 25, 2008 4:30 pm,
Stewart Libary Fields 
B. Valko and B. Virag, University
of Toronto
The Brownian Carousel
The eigenvalues of a random Hermitian matrix form a random
set of points on the real line. As the matrix size converges
to infinity, the eigenvalues, after appropriate scaling, converge
to a point process.
The possible limit processes, called Sinebeta processes, are
fundamental objects of probability theory. They are famous for
their conjectured relationship to the Riemann zeta zeros, Dirichlet
eigenvalues of Euclidean domains, random Young tableaux, and
noncolliding walks.
This series of informal talks is about a new description of
these processes in terms of Brownian motion in the hyperbolic
plane, called the Brownian carousel. We plan to have three lectures:
1. Introduction to random matrix eigenvalues, definition and
basic properties of the Brownian Carousel
2. Computing with the Brownian carousel; continuity, phase transitions,
Dyson's predictions
3. Convergence of finite random matrix eigenvalues to the Brownian
carousel 
Monday,
Feb. 11, 2008 4:30 pm,
Stewart Libary Fields 
Brian Rider (University of Colorado at Boulder)
Diffusion at RMT's hard edge
The RMT hard edge refers to the behavior of the minimal eigenvalues
of a (natural) oneparameter generalization of Gaussian sample
covariance matrices. We show that, in the large dimensional
limit, the law of these points are shared by that of the spectrum
of a certain random secondorderdifferential operator. The
latter may be viewed as
the generator of a Brownian motion with white noise drift.
By a Riccati transform, we get a second diffusion description
of the hard edge in terms of hitting times.
This is joint work with J. Ramirez and should be compared
with slightly less recent results of J. Ramirez, B. Virag,
and myself on the RMT "soft" edge.

Monday,
Feb. 4, 2008 4:10pm,
Stewart Libary Fields 
Omer Angel (University of
Toronto)

Monday,
Dec. 10, 2007 4:10pm,
Stewart Libary Fields 
James Mingo (Queen's University)
Free Cumulants: First and Second Order
Twenty years ago Voiculescu showed that the limiting distribution
of sums and products of some ensembles of random matrices could
be computed using some algebraic methods of "free"
probability. At the core of free probability are the "free"
cumulants. In recent years I have developed with Roland Speicher
a theory of second order cumulants to do for global fluctuations
what Voiculescu's theory did for limiting distributions. 
Monday,
Dec. 3, 2007 4:10pm,
Stewart Libary Fields 
Omer Angel (University of
Toronto)
Minimal Spanning Trees revisited
Given a graph with weighted edges it is easy to find the spanning
tree with minimal total weight. If the graph is the complete
graph K_n and the weights are independent uniform on [0,1] the
MST weight converges in distribution to \zeta(3). I will discuss
two variation on this result.
If the diameter of the tree is constrained to be at most
K, what is the minimal weight? Turns out that there is a transition
at K=\log_2\log n.
If the edges are presented sequentially, and an algorythm
must make a decision on each edge with only partial information,
what can be achieved? Some heuristics lead to algorithms related
to coalescent pocesses. I will give some bounds on the optimal
expected weight.

Monday,
Nov. 26, 2007 4:10pm,
Room 210
Fields 
Balazs Szegedy, University
of Toronto
Forcing Randomness.
A surprising theorem by Chung, Graham and Wilson says that if
a graph has edge density close to 1/2 and four cycle density
close to 1/16 than the structure of the graph is close to "random
looking". The natural question arises: What structures
can be forced upon a graph by a finite family of subgraph densities?
These structures are interesting combinations of algebraic structure
andrandomness. We present recent results in this topic. This
is joint work with Laszlo Lovasz. 
Monday,
Nov. 19, 2007 4:10pm,
Stewart Library
Fields 
Manjunath Krisnapur (University
of Toronto)
From random matrices to random analytic functions.
Peres and Virag proved that the zeros of the power series
a_0+za_1+z^2a_2+..., with i.i.d. standard complex Gaussian coefficients
is a determinantal point process on the unit disk. Extending
this result, I proved recently that the singular points of the
power series A_0+zA_1+z^2A_2+..., where A_i are k x k matrices
with i.i.d. standard complex Gaussian coefficients, is also
determinantal. As this was presented as conjecture in earlier
talks, the emphasis will be on the proof and its connection
to truncations of unitary random matrices sampled according
to Haar measure. 
Monday,
Oct. 29, 2007 4:10pm,
Stewart Library Fields 
Mathieu Merle (University
of British Columbia)
Voter, LotkaVolterra models and superBrownian motion
Voter model was initially interpreted as representing the spread
of an opinion, but as the LotkaVolterra model, it can be also
be interpreted as a stochastic model for competition species.
SuperBrownian motion is a model for population undergoing both
spatial displacement and a continuous branching phenomenon.
Recently, it was shown by Bramson, Cox, Durrett, Le Gall and
Perkins that these objects are closely related, as superBrownian
motion appears at the scaling limit of both voter and LotkaVolterra
models, in dimension greater than two.
Then, know properties of superBrownian motion can be exploited
in order to gain information on these discrete models. We will
see how this leads to asymptotic results for the hitting probabilities
of the voter model started with a single one, in dimensions
2 and 3. We will also briefly survey recent work of Cox and
Perkins, who obtain results on survival and coexistence for
the LotkaVolterra model in dimension greater than 3. 
Monday,
Oct. 15, 2007
4:10pm,
Stewart Library
Fields 
Gidi Amir (University of Toronto)
Excited random walk against a wall
We analyze random walk in the upper half of a three dimensional
lattice which goes down whenever it encounters a new vertex,
reflects on the plane $z=0$, and behaves like a simple random
walk otherwise. a.k.a. excited random walk. We show that it
is recurrent with an expected number of returns of $\sqrt{\log
n}$ (Joint work with Itai Benjamini and Gady Kozma) 
Monday,
Oct. 1, 2007
4:10pm,
Stewart Library
Fields 
Gabor Pete (Microsoft Research)
The exact noise and dynamical sensitivity of critical percolation,
via the Fourier spectrum
Let each site of the triangular lattice (or edge of the \Z^2
lattice) have an independent Poisson clock switching between
open and closed. So, at any given moment, the configuration
is just critical percolation. In particular, the probability
of a leftright open crossing in an n*n box is roughly 1/2,
and, on the infinite lattice, almost surely there are only
finite open clusters.
In the box, how long do we have to wait before we lose essentially
all correlation between having a leftright open crossing
now and then? In the infinite lattice, are there random exceptional
times when there are infinite clusters? In joint work with
Christophe Garban and Oded Schramm, we give quite complete
answers: e.g., exceptional times do exist on both lattices,
and the Hausdorff dimension of their set is computed to be
31/36 for the triangular lattice.
The indicator function of a percolation crossing event is
a function on the hypercube {1,+1}^{sites or edges}, and
thus it has a FourierWalsh expansion. Our proofs are based
on giving sharp estimates on the ``weight'' of the Fourier
coefficients at different frequencies.

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