Introduction to Random Matrices - Theory and Practice

Abstract

This is a book for absolute beginners. If you have heard about random matrix theory, commonly denoted RMT, but you do not know what that is, then welcome!, this is the place for you. Our aim is to provide a truly accessible introductory account of RMT for physicists and mathematicians at the beginning of their research career. We tried to write the sort of text we would have loved to read when we were beginning Ph.D. students ourselves. Our book is structured with light and short chapters, and the style is informal. The calculations we found most instructive are spelt out in full. Particular attention is paid to the numerical verification of most analytical results. Our book covers standard material - classical ensembles, orthogonal polynomial techniques, spectral densities and spacings - but also more advanced and modern topics - replica approach and free probability - that are not normally included in elementary accounts on RMT. This book is dedicated to the fond memory of Oriol Bohigas.

References in corpus (5)

• Path Integral Methods and Applications
• Extreme Eigenvalues of Wishart Matrices: Application to Entangled Bipartite System
• Lessons from Random Matrix Theory for QCD at Finite Density
• Symmetry classes in random matrix theory
• Matrix integrals and enumeration of maps

Cited by in corpus (106)

• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Principal Component Analysis: A Natural Approach to Data Exploration
• The Physics of Financial Networks
• Universality of jamming of non-spherical particles
• Photon transport in a Bose-Hubbard chain of superconducting artificial atoms
• Extreme Decoherence and Quantum Chaos
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Extreme statistics and spacing distribution in a Brownian gas correlated by resetting
• Out-of-time-order correlations and the fine structure of eigenstate thermalisation
• Cavity and replica methods for the spectral density of sparse symmetric random matrices
• Low-frequency vibrational spectrum of mean-field disordered systems
• Wormholes and Spectral Statistics in the Narain Ensemble
• Mean-field model for the density of states of jammed soft spheres
• On matrix models and their $q$-deformations
• Harmonically confined particles with long-range repulsive interactions
• Eigenstate thermalization and rotational invariance in ergodic quantum systems
• Random Matrix Theory of the Isospectral twirling
• Large gaps between random eigenvalues
• Full counting statistics for interacting trapped fermions
• Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems
• Mapping of attention mechanisms to a generalized Potts model
• Classical Glasses, Black Holes, and Strange Quantum Liquids
• Revealing the Micro-Structure of the Giant Component in Random Graph Ensembles
• Emergence of Fermi's Golden Rule
• Full Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems
• Universal non-mean-field scaling in the density of state of amorphous solids
• Optimizing multi-user indoor sound communications with acoustic reconfigurable metasurfaces
• Statistical limits of dictionary learning: random matrix theory and the spectral replica method
• Introduction to correlation networks: Interdisciplinary approaches beyond thresholding
• Eigenvalues of random matrices with generalised correlations: a path integral approach
• Delocalization at small energy for heavy-tailed random matrices
• Capacity-resolution trade-off in the optimal learning of multiple low-dimensional manifolds by attractor neural networks
• Scaling in the eigenvalue fluctuations of the empirical correlation matrices
• The Goldilocks Principle of Learning Unitaries by Interlacing Fixed Operators with Programmable Phase Shifters on a Photonic Chip
• Largest eigenvalue distribution in the double scaling limit of matrix models: A Coulomb fluid approach
• The spectral density of dense random networks and the breakdown of the Wigner semicircle law
• Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties
• Particles confined in arbitrary potentials with a class of finite-ranged interactions
• Scrambling and operator entanglement in local non-Hermitian quantum systems
• Probing quantum chaos in multipartite systems
• Statistically related many-body localization in the one-dimensional anyon Hubbard model
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Jamming and replica symmetry breaking of weakly disordered crystals
• Efficiently Cooling Quantum Systems with Finite Resources: Insights from Thermodynamic Geometry
• Force balance controls the relaxation time of the gradient descent algorithm in the satisfiable phase
• Maximum Entropy approach to multivariate time series randomization
• Inflation as an Information Bottleneck - A strategy for identifying universality classes and making robust predictions
• From entropic to energetic barriers in glassy dynamics: The Barrat-Mézard trap model on sparse networks
• On 1D, N = 4 Supersymmetric SYK-Type Models (I)
• Statistical properties of large data sets with linear latent features
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Decoherence rate in random Lindblad dynamics
• Random features and polynomial rules
• Complex complex landscapes
• Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems
• Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions
• Robustness of Stark many-body localization in the $J_1$-$J_2$ Heisenberg model
• Ground-state energy distribution of disordered many-body quantum systems
• Random Matrix Derived Shrinkage of Spectral Precision Matrices
• Vibrational density of states of jammed packing: mean-field theory
• Effects of phylogeny on coexistence in model communities
• Separability gap and large deviation entanglement criterion
• Resonance distribution in the quantum random Lorentz gas
• Decomposition Pipeline for Large-Scale Portfolio Optimization with Applications to Near-Term Quantum Computing
• Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory
• Gravity.jl: fast and accurate gravitational lens modeling in Julia
• Effects of clustering heterogeneity on the spectral density of sparse networks
• Beyond Fermi's golden rule with the statistical Jacobi approximation
• Statistical Mechanics of Low Angle Grain Boundaries in Two Dimensions
• Annealed averages in spin and matrix models
• Non-Randomness of Google's Quantum Supremacy Benchmark
• Spacings - An Example for Universality in Random Matrix Theory
• Analysis of Bayesian Inference Algorithms by the Dynamical Functional Approach
• Fluctuation corrections to Lifshitz tails in disordered systems
• No-resonance conditions, random matrices, and quantum chaotic models
• On the Spectrum of Multi-Space Euclidean Random Matrices
• Entanglement spectrum of matchgate circuits with universal and non-universal resources
• A memory-based method to select the number of relevant components in Principal Component Analysis
• Non-Markovianity in the Adapted Caldeira-Leggett model
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882-83
• Numerical Simulation of GUE Two-Point Correlation and Cluster Functions
• Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory
• Random matrices associated to Young diagrams
• Spacing Statistics of Energy Spectra: Random Matrices, Black Hole Thermalization, and Echoes
• Extreme-Value Statistics of Stochastic Transport Processes: Applications to Molecular Motors and Sports
• Statistical Mechanics of Dislocation Pileups in Two Dimensions
• Thermalization is typical in large classical and quantum harmonic systems
• Characterizing maximally many-body entangled fermionic states by using $M$-body density matrix
• Universal non-equilibrium dynamics of pure states and density-dependent thermalization in Sachdev-Ye-Kitaev model
• Expansion into the vacuum of stochastic gases with long-range interactions
• Ensemble-averaged mean-field many-body level density: an indicator of integrable versus chaotic single-particle dynamics
• Identifying mobility edge from finite temperature spectral form factor
• Replica approach to the generalized Rosenzweig-Porter model
• Solvable random matrix ensemble with a logarithmic weakly confining potential
• Entanglement Trajectory and its Boundary
• Asymmetry in repeated isotropic rotations
• Magnetic instability and spin-glass order beyond the Anderson-Mott transition in interacting power-law random banded matrix fermions
• Random Matrices and the Free Energy of Ising-Like Models with Disorder
• $\lambda$-shaped random matrices, $\lambda$-plane trees, and $\lambda$-Dyck paths
• Matrix models for extremal and integrated correlators of higher rank
• Random motility regulation as a generic mechanism of community formation
• Spectral properties and coding transitions of Haar-random quantum codes
• Fidelity plateaus from correlated noise in isolated few-level quantum systems
• Proving superintegrability in $\beta$-deformed eigenvalue models
• Localization properties of the sparse Barrat-Mézard trap model