Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions
arXiv:2111.05737
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.
arXiv comment232 pages
References in corpus (45)
- Random-Matrix Theory of Quantum Transport
- The large deviation approach to statistical mechanics
- Level-Spacing Distributions and the Airy Kernel
- Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
- Random Matrix Theory and Chiral Symmetry in QCD
- Random-matrix theory of Majorana fermions and topological superconductors
- Biorthogonal ensembles
- Physics and Mathematics of Calogero particles
- Random matrices and determinantal processes
- Large Deviations of Extreme Eigenvalues of Random Matrices
- 2D growth processes: SLE and Loewner chains
- Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
- Universal singularity at the closure of a gap in a random matrix theory
- Persistence in Nonequilibrium Systems
- 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
- Mastering the Master Field
- Systematic Analytical Approach to Correlation Functions of Resonances in Quantum Chaotic Scattering
- Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
- Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
- Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
- Wigner time delay and related concepts -- Application to transport in coherent conductors
- Invariant -ensembles and the Gauss-Wigner crossover
- Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
- Scaling limit of vicious walks and two-matrix model
- Life and Death in a Cage and at the Edge of a Cliff
- Quantum vs Classical Integrability in Calogero-Moser Systems
- Exact short-time height distribution in 1D KPZ equation with Brownian initial condition
- Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
- On Asymptotics for the Airy Process
- Functional central limit theorems for vicious walkers
- Modelling gap-size distribution of parked cars using random-matrix theory
- Littelmann path model for geometric crystals, Whittaker functions on Lie groups and Brownian motion
- Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
- Quantum Network Models and Classical Localization Problems
- Volumes of orthogonal groups and unitary groups
- Large deviations for extreme eigenvalues of deformed Wigner random matrices
- 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
- Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the -transform
- Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
- 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