Probability 20082009
Probability 20092010

Fall 2011  Spring 2012

September 9, 2011
Phillip Wood
University of Wisconsin (Madison)
Random doubly stochastic tridiagonal matrices
Let T_n be the compact convex set of tridiagonal doubly stochastic
matrices. These arise naturally as birth and death chains with a
uniform stationary distribution. One can think of a "typical" matrix
T_n as one chosen uniformly at random, and this talk will present a
simple algorithm to sample uniformly in T_n. Once we have our hands
on a 'typical' element of T_n, there are many natural questions to
ask: What are the eigenvalues? What is the mixing time? What is the
distribution of the entries? This talk will explore these and other
questions, with a focus on whether a random element of T_n exhibits
a cutoff in its approach to stationarity. Joint work with Persi Diaconis.

September 16, 2011
Carsten Schütt
ChristianAlbrechtsUniversität of Kiel (Germany)
Mahler's conjecture and curvature
Let K be a convex body in R^n with Santalo point at 0. We show that if K has a point on
the boundary with positive generalized Gauss curvature, then the volume product K K^o
is not minimal. This means that a body with minimal volume product has Gauss curvature equal
to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.

September 23, 2011
Prasad Tetali
Georgia Institute of Technology (Atlanta)
Scaling Limits in Sparse Random Graphs via a Combinatorial Interpolation Technique
We establish the existence of free energy limits for several combinatorial
models on ErdosRenyi graph G(N; [cNc]) and random rregular graph G(N;r).
For a variety of models, including independent sets, MAXCUT, Coloring and
KSAT, we prove that the free energy both at a positive and zero temperature,
appropriately rescaled, converges to a limit as the size of the underlying
graph diverges to innity. For example, as a special case we prove that the
size of a largest independent set in these graphs, normalized by the number
of nodes converges to a limit w.h.p. This resolves an open problem which
was proposed by Aldous. It was also mentioned as an open problem in other
works by Wormald, BollobasRiordan, and JansonThomason. Our approach is
based on extending and simplifying the interpolation method of Guerra and
Toninelli, followed by Franz and Leone. We provide a simpler combinatorial
approach and work with the zero temperature case (optimization) directly
both in the case of ErdosRenyi graph G(N;[cN]) and random regular graph
G(N;r). In addition we establish the large deviations principle for the
satisability property of the constraint satisfaction problems such as
Coloring, KSAT and NAEKSAT for the G(N;[Nc]) random graph model.
This is joint work with D. Gamarnik (MIT) and M. Bayati (Stanford).

September 30, 2011
Mokshay Madiman Yale University (New Haven)
Some combinatorial entropy inequalities, with applications
We will provide an overview of a variety of inequalities for discrete
entropy that are indexed by hypergraphs. The first part of the talk (based on
joint work with P. Tetali) will discuss inequalities for entropy of joint
distributions, which generalize various inequalities due to Shannon, Han,
Fujishige and Shearer. The second part of the talk (based on joint work with A.
Marcus and P. Tetali) will discuss inequalities for entropy of sums of
independent random variables, and more generally of socalled
partitiondetermined functions. The latter will be used as tools to develop
general cardinality inequalities for sumsets in possibly nonabelian groups,
with applications to additive combinatorics in mind. In particular,
inequalities that generalize those of GyarmatiMatolcsiRuzsa and
BalisterBollobas will be demonstrated, and partial progress reported towards
a conjecture of Ruzsa for sumsets in nonabelian groups.

October 7, 2011
Frank Aurzada
Technische Universität Berlin (Germany)
The onesided exit problem for integrated processes and fractional Brownian motion
We study the onesided exit problem, also known as onesided
barrier problem, that is, given a stochastic process X we would like to
find, as T\to\infty, the asymptotic rate of
P(sup_{0\leq t\leq T} X_t \leq 1). This question is considered for
alphafractionally integrated centered Levy processes and 'integrated'
centered random walks. The rate of decrease of the above probability is
polynomial with exponent \theta=\theta(\alpha)>0 which only depends on
\alpha but not on the choice of the Levy process or random walk.
This generalizes results of Y.G. Sinai (1991) who considered the simple
random walk and \alpha=1. Similar recent results are due to V. Vysotsky
(2010 and 2011) and A. Dembo and F. Gao (2011). The results are compared
to the corresponding ones for fractional Brownian motion.

October 14, 2011
No seminar this week (Midwest Probability Colloquium)

October 21, 2011
Wenbo Li University of Delaware
Probabilities of all real zeros for random polynomials
There is a long history on the study of zeros of random polynomials whose coefficients
are independent, identically distributed, nondegenerate random variables. We will first provide
an overview on zeros of random functions and then show exact and/or asymptotic bounds on
probabilities that all zeros of a random polynomial are real under various distributions.

October 28, 2011
Elisabeth Werner
Case Western Reserve University (Clevelend)
Functional affineisoperimetry and an inverse logarithmic Sobolev inequality
We give a functional version of the affine isoperimetric inequality
for logconcave functions which may be interpreted as an inverse
form of a logarithmic Sobolev inequality inequality for entropy. A
linearization of this inequality gives an inverse inequality to the
Poincare inequality for the Gaussian measure.

November 4, 2011
Xin Liu IMA, University of Minnesota (Minneapolis)
Diffusion approximations for multiscale stochastic networks in heavy traffic
We study a sequence of nearly critically loaded queueing networks,
with time varying arrival and service rates and routing structure.
The nth network is described in terms of two independent finite state Markov
processes {Xn(t): t = 0} and {Yn(t): t = 0} which can be interpreted as
the random environment in which the system is operating. The process Xn
changes states at a much higher rate than the typical arrival and service times
in the system, while the reverse is true for Yn. The variations in the routing
mechanism of the network are governed by Xn, whereas the arrival and
service rates at various stations depend on the state process
(i.e. queue length process) and both Xn and Yn. Denoting by Zn the suitably
normalized queue length process, it is shown that, under appropriate heavy
traffic conditions, the pair Markov process (Zn, Yn) converges weakly to
the Markov process (Z,Y), where Y is a finite state continuous time Markov
process and the process Z is a reflected diffusion with drift and diffusion
coefficients depending on (Z,Y) and the stationary distribution of Xn.
We also study stability properties of the limit process (Z,Y).

November 11, 2011
Daniel John Fresen University of MissouriColumbia
Concentration inequalities in the space of convex bodies
We discuss the following phenomenon: The convex hull of a large i.i.d. sample from a logconcave
probability measure on Euclidean space approximates a predetermined body in the logarithmicHausdorff
distance (and in the BanachMazur distance). For plogconcave distributions with p>1
(such as the normal distribution where p=2) we also have approximation in the Hausdorff distance.
Three deterministic bodies are given as approximants to the random body; one is the floating body,
another is given as a contour of the density function and the third is given
in terms of the Radon transform. Quantitative bounds are given.

November 18, 2011
Matthew Roberts (Montreal)
A simple path to asymptotics for the frontier of a branching Brownian motion
Bramson's 1978 result on the position of the maximal particle in
a branching Brownian motion has inspired many related results and
generalizations in recent years. We show how modern methods can be used
to give a simpler proof of the original result.

November 25, 2011
No seminar this week (Thanksgiving)

December 2, 2011
Antti Knowles Harvard University
Finiterank deformations of Wigner matrices
The spectral statistics of large Wigner matrices are by now
wellunderstood. They exhibit the striking phenomenon of universality: under
very general assumptions on the matrix entries, the limiting spectral statistics
coincide with those of a Gaussian matrix ensemble. I shall talk about Wigner
matrices that have been deformed by adding a finiterank matrix. By Weyl's
interlacing inequalities, this deformation does not affect the largescale
statistics of the spectrum. However, it may affect eigenvalues near the
spectral edge, causing them to break free from the bulk spectrum.
In a series of seminal papers, Baik, Ben Arous, and Peche (2005) and Peche
(2006) established a sharp phase transition in the statistics of the extremal
eigenvalues of perturbed Gaussian matrices. At the transition, an
eigenvalue detaches itself from the bulk and becomes an outlier.
I shall report on recent joint work with Jun Yin. We consider an NxN Wigner
matrix H perturbed by an arbitrary deterministic finiterank matrix A. We allow
the eigenvalues of A to depend on N. Under optimal (up to factors of log N)
conditions on the eigenvalues of A, we identify the limiting distribution of
the outliers of H+A. We also prove that the remaining eigenvalues of H+A "stick"
to eigenvalues of H, thus establishing the edge universality of H+A. On the
other hand, our results show that the distribution of the outliers is not
universal, but depends on the distribution of H and on the geometry of the
eigenvectors of A. As the outliers approach the bulk spectrum, this dependence
is washed out and the distribution of the outliers becomes universal.

December 9, 2011
Michael Damron Princeton University
A simplified proof of the relation between scaling exponents in
firstpassage percolation
In first passage percolation, we place i.i.d. nonnegative weights
on the nearestneighbor edges of Z^d and study the induced random metric.
A longstanding conjecture gives a relation between two "scaling exponents":
one describes the variance of the distance between two points and the other
describes the transversal fluctuations of optimizing paths between the same
points. This is sometimes referred to as the "KPZ relation."
In a recent breakthrough work, Sourav Chatterjee proved this conjecture using
a strong definition of the exponents. I will discuss work I just completed
with Tuca Auffinger, in which we introduce a new and intuitive idea that
replaces Chatterjee's main argument and gives an alternative proof of the
relation. One advantage of our argument is that it does not require a certain
nontrivial technical assumption of Chatterjee on the weight distribution.

December 16, 2011  January 7, 2012
No seminar this period

January 13, 2012
WeiKuo Chen Department of Mathematics, UC Irvine
The Chaos problem in the SherringtonKirkpatrick model
In physics, the main objective in spin glasses is to understand
the strange magnetic properties of alloys. Yet the models invented to
explain the observed phenomena are of a rather fundamental nature in
mathematics. In this talk, we will focus on one of the most important mean
field models, called the SherringtonKirkpatrick model, and discuss its
disorder and temperature chaos problems. Using the Guerra replica
symmetricbreaking bound and GhirlandaGuerra identities, we will present
mathematically rigorous results and proofs for these problems.

January 20, 2012
Arnab Sen University of Cambridge (UK)
Random Toeplitz matrices
Random Toeplitz matrices belong to the exciting area that lies at the
intersection of the usual Wigner random matrices and random Schrodinger
operators. In this talk I will describe two recent results on random Toeplitz
matrices. First, the maximum eigenvalue, suitably normalized, converges to
the 24 operator norm of the wellknown Sine kernel. Second, the limiting
eigenvalue distribution is absolutely continuous, which partially settles
a conjecture made by Bryc, Dembo and Jiang (2006). I will also present
several open questions and conjectures towards the end of the talk.
This is joint work with Balint Virag (Toronto).

January 27, 2012
Xiaoqin Guo University of Minnesota
Einstein relation for random walks in random environment
In this talk, we will prove the Einstein relation for random walks in
a uniformly elliptic and iid random environment. We consider the velocity of
the random walks in a perturbation of a balanced random environment
(i.e., environment with zero drift). We will show that the ratio between
the velocity and the size of the perturbation converges to a diffusivity
constant of the balanced environment.
This talk is based on a joint work with my advisor Ofer Zeitouni.

February 3, 2012
Dmitriy Bilyk University of Minnesota
Small ball probabilities for the Brownian sheet, uniform
distributions, and related problems
We shall discuss recent progress on the asymptotic estimates for
multiparameter Gaussian processes, such as the Brownian sheet. As
shown by Kuelbs and Li, 1993, these inequalities are equivalent to the
entropy estimates of Sobolev spaces with mixed smoothness. The methods
of obtaining lower bounds are rooted in the work of Talagrand, 1994,
and connect this problem to interesting questions in multiparameter
wavelet analysis. Higher dimensional analogues of Talagrand's theorem
turned out to be extremely proofresistant, and only partial results
are availble in dimensions three and higher (Beck, 1989; Bilyk, Lacey,
Vagharshakyan, 200810).
This problem is also connected to the grand open problem of the theory
of irregularities of distribution  the exact rate of growth of the
discrepancy function. We shall examine this relation as well as the
use of probabilistic methods in numerous other questions of
discrepancy theory.

February 10, 2012
Krzysztof Burdzy
University of Washington (Seattle)
Deterministic approximations of random reflectors
Every random reflector on a line which satisfies a natural condition
(well established in the theory of billiards) can be approximated by a
sequence of deterministic reflectors represented by families of
mirrors. Joint work with O. Angel and S. Sheffield.

February 17, 2012
Nick Krylov University of Minnesota
Accelerated finitedifference schemes for linear stochastic partial
differential equations in the whole space (Joint with Istv\'an Gy\"ongy)
We give sufficient conditions under which the convergence of finite difference
approximations in the space variable of the solution to the Cauchy problem
for linear stochastic PDEs of parabolic type can be accelerated to any
given order of convergence by Richardson's method.

February 24, 2012
John Baxter University of Minnesota
Random Equilibrium Measures

March 2, 2012
Rafail Khasminskii Wayne State University (emeritus)
Stability of Stochastic Differential Equations with Markovian Switching
The main aim of this talk is to present a result
concerning reducing asymptotic stability problem for stochastic
differential equation (SDE) with sufficiently rapid Markovian
switching to the wellstudied analogous problem for the "averaged" SDE
without switching. Application to the problem of stabilization by
switching and to ordinary differential equation (ODE) with switching
will be also presented.

March 9, 2012
Sergey Bobkov University of Minnesota
Rate of convergence in the entropic central limit theorem
For the scheme of i.i.d. random variables, we discuss rates of convergence of entropy of
the normalized sums to the entropy of gaussian limit.
Joint work with G. P. Chistyakov and F. Götze.

Mach 16, 2012
No seminar this week (Spring Break)

March 23, 2012
Greg Anderson University of Minnesota
Joint cumulants of functions of a Gaussian random vector and
freeprobabilistic counting of colored planar maps
Our main result is an explicit operatortheoretic formula for the number of
colored planar maps with a fixed set of stars each of which has a fixed set
of halfedges with fixed coloration. The formula transparently bounds
the number of such colored planar maps and does so well enough to prove
convergence near the origin of generating functions arising naturally in
the matrix model context. Such convergence is known but the proof of
convergence proceeding by way of our main result is relatively simple.
Our main technical tool is an integration identity representing the joint
cumulant of several functions of a Gaussian random vector. In the case
of cumulants of order 2 the identity reduces to one wellknown as a means
to prove the Poincare inequality.

March 30, 2012
Wenbo Li University of Delaware and IMA
Small value probabilities and branching related processes
We first provide an overview on fundamental roles of small value
probability (estimates of rare events that positive random variables take
smaller values than typical ones) in the theory of stochastic processes.
Then we focus on estimates associated with variants of Branching processes
and their martingale limits. Relevant techniques and tools will be discussed
in the simplest setting.

April 6, 2012
Tiefeng Jiang
School of Statistics, University of Minnesota
Distributions of Angles in Random Packing on Spheres
We study the asymptotic behaviors of the pairwise angles among n randomly and uniformly
distributed unit vectors in pdimensional spaces as the number of points n goes to infinity,
while the dimension p is either fixed or growing with n. For both settings, we derive
the limiting empirical distribution of the random angles and the limiting distributions of
the extreme angles. The results reveal interesting differences in the two settings and
provide a precise characterization of the folklore that "all highdimensional random vectors
are almost always nearly orthogonal to each other". Applications to statistics and connections
with some open problems in physics and mathematics are also discussed.
This is a joint work with Tony Cai and Jianqing Fan.

April 13, 2012
Bert Fristedt
University of Minnesota
Random sets and random discrete distributions
Let Z be a random closed subset of the nonnegative reals. Suppose that it has measure 0 and
that it is selfsimilar in the sense that cZ has the same distribution as Z for all positive
constants c. The set [0,1]\Z is an open set with measure 1. The distribution of Z is
characterized by the distribution of the length of the rightmost open interval in [0,1]\Z.
This distribution has a density f(u) satisfying: (1u)f(u) is a decreasing function of u.
Moreover, any such distribution on (0,1) can arise in this manner. Two examples of Z are:
(1) the set of zeroes of Brownian motion; (2) The set of values of a Levy process with
Levy measure x^{1} dx. The main reference is a paper by Jim Pitman and Marc Yor
entitled "Random Discrete Distributions Derived from SelfSimilar Random Sets"
which appeared in The Electronic Journal of Probability, 1996, Vol. 1, pages 128.

April 20, 2012
No seminar this week (RivièreFabes Symposium)

April 27, 2012
Maury Bramson
University of Minnesota
Pursuit Games and Shy Couplings. Part I
We are interested in the question of when a shy coupling, for a pair of Brownian motions,
exists in a given bounded domain. That is, is it possible to construct a (nonanticipating)
coupling between a pair of Brownian motions so that, with positive probability, they are
always at least some assigned distance apart? This question is related to the
deterministic Lion and Man problem, with the Lion attempting to capture the Man when
both are allowed to move at the same rate. This talk is based on joint work with
K. Burdzy and W. Kendall.

May 4, 2012
Maury Bramson
University of Minnesota
Pursuit Games and Shy Couplings. Part II
This is a continuation of Part I.
