Two problems in Probability

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Under the name of ``spin glass models" physicists have studied by non-rigorous methods a number of fundamental mathematical objects. These are rather canonical very large families of correlated random variables with a smooth ``high-dimensional" correlation structure, which are very different from the structures investigated by classical probability theory. The physicists predict the emergence under very general conditions of a kind of self-organization called ultrametricity. We suspect that the root of this phenomenon lies in a family of probabilistic identities discovered a few years ago by Ghirlanda and Guerra and we plan to investigate whether this is really the case. A somewhat different direction of the proposal plans to focus on the general theory of stochastic processes. There are indications that some new structure theorems are possible, that would in some precise sense affirm that if certain types of stochastic processes are well-behaved according to their sample boundedness properties, then there is simple ``witness'' of their good behavior. This would imply that when trying to decide whether a specific stochastic process has a good behavior, this can be done only by finding the appropriate witness and by no other means. Both part of the proposal represent a genuinely new direction of Probability theory. A growing number of large scale operations (safety of electrical networks, safety of nuclear plants, finance) essentially depend on tools from Probability theory. This proposal investigates two new directions in this theory. The first concerns the emergence in certain large random structures of remarkable kind of self-organization discovered by theoretical physicists. The second attempts to prove that there is a kind of universal method to control how large can be the ``worst occurrence'' of certain random quantities, and that therefore it is useless to attempt to do this in any other way than as prescribed by the universal method.