Title: Disordered statistical physics in low dimensions: extremes, glass transition, and localization,
Abstract: This thesis presents original results in two domains of disordered
statistical physics: logarithmic correlated Random Energy Models (logREMs), and
localization transitions in long-range random matrices.
In the first part devoted to logREMs, we show how to characterise their
common properties and model--specific data. Then we develop their replica
symmetry breaking treatment, which leads to the freezing scenario of their free
energy distribution and the general description of their minima process, in
terms of decorated Poisson point process. We also report a series of new
applications of the Jack polynomials in the exact predictions of some
observables in the circular model and its variants. Finally, we present the
recent progress on the exact connection between logREMs and the Liouville
conformal field theory.
The goal of the second part is to introduce and study a new class of banded
random matrices, the broadly distributed class, which is characterid an
effective sparseness. We will first study a specific model of the class, the
Beta Banded random matrices, inspired by an exact mapping to a recently studied
statistical model of long--range first--passage percolation/epidemics dynamics.
Using analytical arguments based on the mapping and numerics, we show the
existence of localization transitions with mobility edges in the
"stretch--exponential" parameter--regime of the statistical models. Then, using
a block--diagonalization renormalization approach, we argue that such
localization transitions occur generically in the broadly distributed class.