
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

201314
Toronto Probability
Seminar
held
at the Monday 3 p.m. in Room 210
For
questions, scheduling, or to be added to the
mailing list, contact the organizers at: probsem<@>math.toronto.ed


Organizers
Bálint
Virág , Janosch
Ortmann University of Toronto, Mathematics
and Statistics



Mondays 
Upcoming
Seminars at 3:10 p.m. in the Fields Institute, Room 210


TBA

Past Talks 
January 17 at
2:10 p.m.
Stewart Library 
Balint Virag
Covariance structures for iid factors on regular trees
There is a oneparameter family of 01 valued Markov processes indexed
by the dregular tree. We would like to construct these processes
as an invariant, deterministic function (factor) of iid random variables
on the vertices. It is an open problem to determine when this is possible.
I will review how this is connected to independent sets, max and
min cut problems on large girth graphs, and BenjanminiSchramm convergence.
While it is still mystery what processes can be a factor of iid, a
lot more is known about when a covariance structure has this property.
Join work with Agnes Backhausz and Balazs Szegedy.

January 13

Lerna Pehlivan (Washington)
Structure of Random 312Avoiding Permutations
A permutation of {1,2,..,N} is said to avoid 312 pattern if there
is no subsequence of three elements of this permutation that appears
at the same relative order as 312. Monte Carlo experiments reveal
some features of random 312 avoiding permutations. In light of these
experiments we determine some probabilities explicitly.
This paper is a joint work with Neal Madras.

Nov. 25

Andrew Stewart (Toronto)
The scaling limit of the range of the simple random walk bridge
on regular trees
We consider the lazy simple random walk bridge of length n on a dregular
tree. The range of the simple random walk bridge, the set of vertices
$R_n$ visited by a bridge of length $n$, is a finite tree whose diameter
is $\approx \sqrt{n}$. We show that the metric space $R_n/\sqrt{n}$
converges in distribution in the GromovHausdorf metric to the Brownian
Continuum Random Tree introduced by Aldous. We use techniques introduced
in [1].
This is joint work with Balint Virag.
[1] Aldous, D. The continuum random tree III. Ann. Probab. Volume
21,
Number 1 (1993), 248289.

Nov. 20
BA 6183 at 5.10 pm 
Special Probability Seminar
Werner Kirsch (Hagen).
Spectral Theory for Block Matrices with Random Entries
We discuss spectral properties of some block matrices whose entries
are random Schroedinger operators. These operators model certain systems
connected to superconductor physics. We will concentrate on matrices
in the BCS form which arises in the theory of
superconductors. We concentrate on properties of the density of states
for those operators as well as on the question of Anderson localization.

Nov. 4 
Tom Bloom (Toronto)
Large deviation for outlying coordinates in ß ensembles
Abstract: ß ensembles are generalizations of the joint probability
distribution of the eigenvalues of the GOE and GUE. A ß ensemble on
a compact set K?C is a probability distribution Prob_{n,ß} on K_n
for n=1,2...
We show that the related sequence of probability distributions on
K defined by, for W $?$ K, Prob_{n,ß}(z1$ ? $W) where z1 is the first
coordinate, satisfies a large deviation principle with speed n and
an explicit rate function. This extends work of BorotGuionnet from
ß ensembles on R.

October 28 
Almut Burchard (Toronto)
Some applications of twopoint symmetrization in Probability
Twopoint symmetrization is a simple equimeasurable rearrangement
of sets and functions that pushes mass towards the origin. It is often
used to prove that certain symmetric functionals have radially symmetric
extremals; it can be particularly helpful for identifying equality
cases. I will describe a rearrangement inequality for multiple integrals,
and discuss classical and recent applications to geometric problems
involving path integrals.

October 21 
Ben Rifkind
Eigenvectors of the 1D Random Schrodinger Operator
We consider a model of the one dimensional discrete random Schrodinger
operator on Z_n given by H_n = L_n + V_n, where L_n is the discrete
Laplacian and V_n is a random potential. If v_k := (V_n)_{kk} does
not depend on n, the eigenvectors are localized (Carmona et al., 1987)
and the local statistics of eigenvalues are Poisson. In order to capture
the transition between localization and delocalization Kritchevski,
Valko, and Virag (2011) analyzed the model in the case when v_k decays
like n^(1/2) and characterized the local statistics of eigenvalues.
Building from the framework developed in that paper, I will discuss
scaling limits of the corresponding eigenvectors. They converge (in
some sense) to a simple function of Brownian motion. This is joint
work with Balint Virag.

October 7 
Van Vu
How many real roots does a random polynomial have?
Consider a polynomial P_n = c_0 + c1x +...c_n x^n of degree n whose
coefficients c_i are (not necessarily iid) real random variables.
The problem of determining N_n, the number of real zeroesof P_n goes
back to Waring (1782), and has become popular since the series of
works of Littlewood and Offord in the 1940s. Deep works of LittlewoodOfford,
ErdosOfford, Turan, Kac, Stevens, IbragimovMaslova, EdelmanKostlan
and many others give us a good understanding of N_n in the case c_i
are iid random variables with mean 0 and variance 1 (see John Baez's
"Lord of the Ring" beautiful picture on http://math.ucr.edu/home/baez/).
However, much less is known for the all other cases, when the c_i
may have different variances and/or are dependent (a good example
is the characteristic polynomial of a random matrix).
In this talk, I am going to give a brief survey on the state of the
art of the problem, and introduce a new approach, developed together
with T. Tao, that leads to a very good understanding of real (and
also complex) roots of a very general class of random polynomials.

September 30 
Mustazee Rahman
Independent sets in random regular graphs
An independent set in a graph is a set of vertices such that there
are no edges between them. How large can an independent set be in
a random dregular graph? How large can it be if we are to construct
it using a (possibly randomized) algorithm that is local in nature?
The talk will disuss a recently introduced notion of local algorithms
for combinatorial optimization problems on large, random dregular
graphs. The talk will then explain why, for asymptotically large d,
local algorithms can only produce independent sets of size at most
half of the largest ones. The factor of 1/2 turns out to be optimal.
Joint work with Balint Virag.

September 23 
Janosch Ortmann
The KPZ equation and interacting particle systems
The KhardarParisiZhang (KPZ) equation is a stochastic partial differential
equation used to describe randomly evolving interfaces. Its solution
has an unusual scaling behaviour, and the distribution of the fluctuations
are related to random matrices. The class of such models is called
the KPZ universality class and is predicted to contain a number of
discrete and semidiscrete models. We will discuss some of these models
and recent progress made towards establishing this universality for
the socalled halfflat and flat initial conditions for the Asymmetric
Simple Exclusion Process, based on joint work with Jeremy Quastel
and Daniel Remenik.



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