Georgia Institute of Technology College of Sciences

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.

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.

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing onthe exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads tosharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinations of various procedures such as least square regression,kernel ridge regression, shrinking estimators and many other estimators used in the literatureon statistical inverse problems. As a consequence, we show that the proposed aggregate provides an adaptive estimator in the exact minimax sense without neither discretizing the rangeof tuning parameters nor splitting the set of observations. We also illustrate numerically thegood performance achieved by the exponentially weighted aggregate. (This is a joint work with Arnak Dalalyan.)

In this talk I will discuss random matrices that are matricial analogs of the well known binomial, Poisson, and negative binomial random variables. The common thread is the conditional variance of X given S = X+X', which is a quadratic polynomial in S and in the univariate case describes the family of six Meixner laws that will be described in the talk. The Laplace transform of a general n by n Meixner matrix ensemble satisfies a system of PDEs which is explicitly solvable for n = 2. The solutions lead to a family of six non-trivial 2 by 2 Meixner matrix ensembles. Constructions for the "elliptic cases" generalize to n by n matrices. The talk is based on joint work with Gerard Letac.