PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"

Abstract

In this thesis, we study three physically relevant models of strongly correlated random variables: trapped fermions, random matrices and random walks. In the first part, we show several exact mappings between the ground state of a trapped Fermi gas and ensembles of random matrix theory. The Fermi gas is inhomogeneous in the trapping potential and in particular there is a finite edge beyond which its density vanishes. Going beyond standard semi-classical techniques (such as local density approximation), we develop a precise description of the spatial statistics close to the edge. This description holds for a large universality class of hard edge potentials. We apply these results to compute the statistics of the position of the fermion the farthest away from the centre of the trap, the number of fermions in a given domain (full counting statistics) and the related bipartite entanglement entropy. Our analysis also provides solutions to open problems of extreme value statistics in random matrix theory. We obtain for instance a complete description of the fluctuations of the largest eigenvalue in the complex Ginibre ensemble. In the second part of the thesis, we study extreme value questions for random walks. We consider the gap statistics, which requires to take explicitly into account the discreteness of the process. This question cannot be solved using the convergence of the process to its continuous counterpart, the Brownian motion. We obtain explicit analytical results for the gap statistics of the walk with a Laplace distribution of jumps and provide numerical evidence suggesting the universality of these results.

212 pages, 66 figures

References in corpus (30)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• A Quantum Gas Microscope for Fermionic Atoms
• Quantum Noise as an Entanglement Meter
• Growing interfaces uncover universal fluctuations behind scale invariance
• Homogeneous Atomic Fermi Gases
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Full counting statistics in a propagating quantum front and random matrix spectra
• Entanglement from Charge Statistics: Exact Relations for Many-Body Systems
• The entanglement entropy of one-dimensional gases
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Universal Record Statistics of Random Walks and Lévy Flights
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Exact distribution of the maximal height of p vicious walkers
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Random matrices and entanglement entropy of trapped Fermi gases
• Record statistics for biased random walks, with an application to financial data
• Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
• Record statistics and persistence for a random walk with a drift
• Generalized arcsine laws for fractional Brownian motion
• Density of near-extreme events
• Record Statistics for Multiple Random Walks
• Universal Order and Gap Statistics of Critical Branching Brownian Motion
• Branching Brownian Motion Conditioned on Particle Numbers
• Breaking supersymmetry in a one-dimensional random Hamiltonian
• Extreme value statistics of correlated random variables