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

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.

232 pages

References in corpus (18)

• The large deviation approach to statistical mechanics
• Physics and Mathematics of Calogero particles
• Large Deviations of Extreme Eigenvalues of Random Matrices
• 2D growth processes: SLE and Loewner chains
• The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Exact short-time height distribution in 1D KPZ equation with Brownian initial condition
• Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
• Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion
• Survival of a diffusing particle in an expanding cage
• Expansion of a matter wave packet in a one-dimensional disordered potential in the presence of a uniform bias force
• Rank one HCIZ at high temperature: interpolating between classical and free convolutions
• Long-range interactions in the avalanches of elastic interfaces
• From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models
• Mean-field methods and algorithmic perspectives for high-dimensional machine learning

Cited by in corpus (1)

• Conditioning two diffusion processes with respect to their first-encounter properties