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.

Ph.D. thesis, 154 pages, 47 figures

References in corpus (20)

• Anderson Transitions
• Localization of interacting fermions at high temperature
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Fractional Quantum Hall States and Jack Polynomials
• Exact relations between multifractal exponents at the Anderson transition
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Non-ergodic phases in strongly disordered random regular graphs
• The acceleration of evolutionary spread by long-range dispersal
• On the scaling of the chemical distance in long-range percolation models
• Many-Body Delocalization in Strongly Disordered System with Long-Range Interactions: Finite Size Scaling
• Distribution functions for largest eigenvalues and their applications
• Mean-field avalanches in jammed spheres
• Duality and KPZ in Liouville Quantum Gravity
• Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
• Freezing and decorated Poisson point processes
• Difference between level statistics, ergodicity and localization transitions on the Bethe lattice
• Multifractal states in self-consistent theory of localization: analytical solution
• Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together
• Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
• On the construction of the correlation numbers in Minimal Liouville Gravity