Random matrices, complexity of spin glasses and heavy tailed processes.

摘要

In the first part of this thesis we introduce a new identity relating critical values of random Hamiltonians in certain compact manifolds to eigenvalues of random matrix ensembles. This identity allows us identify the location of the ground state energy and obtain an explicit formula for the asymptotic complexity of the number of critical points of finite and diverging index at any level of energy. We show that two possible scenarios for the bottom energy landscape emerge. This picture is consistent with a transition from a glass to spin glass system.;In the second part, we establish the limit laws for largest eigenvalues of Wigner and Sample Covariance matrices when the entries are heavy tailed with less than four moments. We then study a model of Directed Polymer in a heavy tailed Random Environment proving a scaling limit law for the polymer measure.