Probability and ergodic theory (PET) Ical Atom

The Hamiltonian of the spherical p-spin spin glass model is a smooth Gaussian field on the N-dimensional sphere. Let $Crt_N(u)$ denote the number of its critical points below $Nu$. In a recent study Auffinger, Ben Arous, and Cerny computed the mean of $Crt_N(u)$ and its exponential growth rate, as N goes to infinity. Our work focuses on the computation of the second moment. We prove that the ratio of second to first moment squared goes to 1, as N goes to infinity. An immediate consequence of this is that $Crt_N(u)$ concentrates around its mean: $Crt_N(u)$ normalized by its mean goes to 1 in L^2 and thus in probability. Joint work with Ofer Zeitouni.

We shall discuss connections between a statistical model for learning (probably approximately correct) and the ability to compress information (sample compression schemes). Joint work with Shay Moran.

Consider a random finite metric space X given by sampling n points in the unit interval uniformly, and a deterministic finite metric space U given by placing n points in the unit interval at uniform distance. With high probability, X will contain some pairs of points at distance roughly 1/n^2, so any bijection from X to U must distort distances by a factor of roughly n. However, with high probability, two of these random spaces, X_1 and X_2, have a bijection which distorts distances by a factor of only about n^(2/3). The exponent of 2/3 is optimal.

The Stochastic Transport Equation describes the number distribution of a population governed by a birth, death and branching event rates, often referred to as a ”Continuous time Markovian branching process”. Continuous time branching processes are a common model of the neutron population in a fissionable system. In particular, the stochastic transport equation is often used in the context of the so called Feynman - alpha method, here the first two moments are used to evaluate the decay rate of the system. In the study, we have extended the traditional model into a multi type setting. In particular, we have demonstrated that the classic results have a very elegant Matricidal representation, if the proper formalism is used.