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Series: Stochastics Seminar

In this talk we will describe the different behaviors of solutions of the random Schrödinger with long-range correlations. While in the case of arandom potential with rapidly decaying correlations nontrivial phenomenaappear on the same scale, different phenomena appear on different scalesfor a random potential with slowly decaying correlations nontrivial .

Series: Stochastics Seminar

We review various classifications of probability
distributions based on their tail heaviness. Using
a characterization of medium-tailed distributions
we propose a test for testing the null hypothesis
of medium-tail vs long- or short-tailed distributions.
Some operating characteristics of the proposed
test are discussed.

Series: Stochastics Seminar

The marginal distribution of identically distributed random variables
having a constant sum is called a completely mixable distribution. In
this talk, the concept, history and present research of the complete
mixability will be introduced. I will discuss its relevance to existing
problems in the Frechet class, i.e. problems with known marginal
distributions but unknown joint distribution and its applications in
quantitative risk management.

Series: Stochastics Seminar

Excursion sets of stationary random fields have attracted much attention in recent years.They have been applied to modeling complex geometrical structures in tomography, astro-physics and hydrodynamics. Given a random field and a specified level, it is natural to studygeometrical functionals of excursion sets considered in some bounded observation window.Main examples of such functionals are the volume, the surface area and the Euler charac-teristics. Starting from the classical Rice formula (1945), many results concerning calculationof moments of these geometrical functionals have been proven. There are much less resultsconcerning the asymptotic behavior (as the window size grows to infinity), as random variablesconsidered here depend non-smoothly on the realizations of the random field. In the talk wediscuss several recent achievements in this domain, concentrating on asymptotic normality andfunctional central limit theorems.

Series: Stochastics Seminar

We consider the problem of estimating the covariance matrix. Factormodels and random effect models have been shown to provide goodapproximations in modeling multivariate observations in many settings. These models motivate us to consider a general framework of covariancestructures, which contains sparse and low rank components. We propose aconvex optimization criterion, and the resulting estimator is shown torecover exactly the rank and support of the low rank and sparsecomponents respectively. The convergence rates are also presented. Tosolve the optimization problem, we propose an iterative algorithm basedon Nesterov's method, and it converges to the optimal with order 1/t2for any finite t iterations. Numerical performance is demonstratedusing simulated data and stock portfolio selection on S&P 100.(This is joint work with T. Tony Cai.)

Series: Stochastics Seminar

In work on the Riemann zeta function, it is of interest to
evaluate certain integrals involving the characteristic polynomials of
N x N unitary matrices and to derive asymptotic expansions of
these integrals as N -> \infty. In this talk, I will obtain exact
formulas for several of these integrals, and relate these results to
conjectures about the distribution of the zeros of the Riemann zeta
function on the critical line. I will also explain how these results are
related to multivariate statistical analysis and to the hypergeometric
functions of Hermitian matrix argument.

Series: Stochastics Seminar

I will talk briefly some of my recent research on random networks. In the
first part of the talk, we will focus on the connectivity of a random
network. The network is formed from a set of randomly located points and
their connections depend on the distance between the points. It is clear
that the probability of connection depends on the density of the points.
We will explore some properties of this probability as a function of the
point density. In the second part, I will discuss a possible approach in
the study correlation structure of a large number of random variables. We
will focus mainly on Gaussian distribution and distributions which are
"similar" to Gaussian distributions. The idea is to use a single number to
quantify the strength of correlation among all the random variables. Such
a quantity can be derived from a latent cluster structure within a
Markovian random network setting.

Series: Stochastics Seminar

In this talk, we resolve several questions related to a certain heavy
traffic scaling regime (Halfin-Whitt) for parallel server queues, a family
of stochastic models which arise in the analysis of service systems. In
particular, we show that the steady-state queue length scales like
$O(\sqrt{n})$, and bound the large deviations behavior of the limiting
steady-state queue length. We prove that our bounds are tight for the
case of Poisson arrivals. We also derive the first non-trivial bounds for
the steady-state probability that an arriving customer has to wait for
service under this scaling. Our bounds are of a structural nature, hold
for all $n$ and all times $t \geq 0$, and have intuitive closed-form
representations as the suprema of certain natural processes. Our upper
and lower bounds also exhibit a certain duality relationship, and
exemplify a general methodology which may be useful for analyzing a
variety of stochastic models. The first part of the talk is joint work
with David Gamarnik.

Series: Stochastics Seminar

The Burgers equation is a nonlinear PDE and one of the basic
hydrodynamic models. The ergodic theory of the Burgers turbulence began with
the work of E, Khanin, Mazel, Sinai (Ann. Math. 2000). In their paper and in
subsequent papers by Khanin and his coauthors, the compact case (Burgers on
a circle or torus) was studied. In this talk, I will discuss the noncompact
case. The main object is optimal paths through clouds of Poissonian points.

Series: Stochastics Seminar

The Potts antiferromagnet on a random graph is a model
problem from disordered systems, statistical mechanics with random
Hamiltonians. Bayati, Gamarnik and Tetali showed that the free energy
exists in the thermodynamic limit, and demonstrated the applicability
of an interpolation method similar to one used by Guerra and
Toninelli, and Franz and Leone for spin glasses. With Contucci,
Dommers and Giardina, we applied interpolation to find one-sided
bounds for the free energy using the physicists' ``replica symmetric
ansatz.'' We also showed that for sufficiently high temperatures, this
ansatz is correct. I will describe these results and some open
questions which may also be susceptible to the interpolation method.