Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions

TL;DR

This thesis explores the connections between random matrix eigenvalues, non-crossing random walks, and trapped fermions, using tools from random matrix theory, stochastic calculus, and quantum mechanics to address original problems and reveal new links.

Contribution

It introduces novel links between inverse-Wishart ensembles and trapped fermions, and generalizes the Ferrari-Spohn problem for non-crossing scalar bridges.

Findings

Established deep links between eigenvalues of random matrices and non-crossing random walks.

Linked inverse-Wishart ensemble to fermions in Morse potential.

Generalized Ferrari-Spohn problem for non-crossing scalar bridges.

Abstract

This statistical physics thesis focuses on the study of three kinds of systems which display repulsive interactions: eigenvalues of random matrices, non-crossing random walks and trapped fermions. These systems share many links, which can be exhibited not only at the level of their static version, but also at the level of their dynamical version. We present a combined analysis of these systems, employing tools of random matrix theory and stochastic calculus as well as tools of quantum mechanics, in order to solve some original problems. Further from the detailed presentation of the field and the report of the results obtained during the PhD, the different themes exposed in the chapters of the thesis allow for perspectives on related issues. As such, the first chapter is an introduction to random matrix theory; we detail its historical evolution and numerous applications, and present its essential concepts, constructions and results. The second chapter discusses non-crossing random walks; we describe the deep links they share with random matrix eigenvalue processes and showcase the results obtained in the scope of boundary problems. In the third chapter, which focuses on stochastic matrix processes, we introduce in particular a process inspired from the Kesten random recursion, and highlight the new link it allows to draw between the inverse-Wishart ensemble and fermions trapped in the Morse potential. Lastly, the fourth chapter, centred on the particular case of bridge processes, allows for a joint treatment of scalar and matrix models; therein, we develop a generalization of the Ferrari-Spohn problem for non-crossing scalar bridges and, as an opening, we exhibit the connections of matrix bridges with other aspects of random matrices.