Amir Dembo

Information theoretic inequalities

The role of inequalities in information theory is reviewed, and the relationship of these inequalities to inequalities in other branches of mathematics is developed. The simple inequalities for differential entropy are applied to the standard multivariate normal to furnish new and simpler proofs of the major determinant inequalities in classical mathematics. The authors discuss differential entropy inequalities for random subsets of samples. These inequalities when specialized to multivariate normal variables provide the determinant inequalities that are presented. The authors focus on the entropy power inequality (including the related Brunn-Minkowski, Youngs, and Fisher information inequalities) and address various uncertainty principles and their interrelations. >

High-order absolutely stable neural networks

The stability properties of arbitrary order continuous-time dynamic neural networks are studied in the spirit of an earlier analysis of a first-order system by M.A. Cohen and S. Grossberg (1983). The corresponding class of Lyapunov function is presented and the equilibrium points are characterized. The relationships with other continuous-time models are pointed out. >

Aging of spherical spin glasses

Abstract. Sompolinski and Zippelius (1981) propose the study of dynamical systems whose invariant measures are the Gibbs measures for (hard to analyze) statistical physics models of interest. In the course of doing so, physicists often report of an “aging” phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified spherical version of this model. The induced rotational symmetry of the spherical model reduces the dynamics in question to an N-dimensional coupled system of Ornstein-Uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices. We obtain the limiting dynamics for N approaching infinity and by analyzing its long time behavior, explain what is aging (mathematically speaking), what causes this phenomenon, and what is its relationship with the phase transition of the corresponding equilibrium invariant measures.

Stochastic Interface Models

In these notes we try to review developments in the last decade of the theory on stochastic models for interfaces arising in two phase system, mostly on the so-called ⊸φ interface model. We are, in particular, interested in the scaling limits which pass from the microscopic models to macroscopic level. Such limit procedures are formulated as classical limit theorems in probability theory such as the law of large numbers, the central limit theorem and the large deviation principles.

Large Deviations for Quadratic Functionals of Gaussian Processes

AbstractThe Large Deviation Principle (LDP) is derived for several quadratic additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to

Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk

1/T\int {_0^T } X_t^2 dt

Embedding nonnegative definite Toeplitz matrices in nonnegative definite circulant matrices, with application to covariance estimation

is the Fenchel-Legendre transform of