Distribution functions for largest eigenvalues and their applications

Abstract

It is now believed that the limiting distribution function of the largest eigenvalue in the three classic random matrix models GOE, GUE and GSE describe new universal limit laws for a wide variety of processes arising in mathematical physics and interacting particle systems. These distribution functions, expressed in terms of a certain Painlevé II function, are described and their occurences surveyed.

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• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• The Pearcey Process
• Random matrix techniques in quantum information theory
• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Functionals of the Brownian motion, localization and metric graphs
• Slow decorrelations in KPZ growth
• Does the SYK model have a spin glass phase?
• Nuclei, Primes and the Random Matrix Connection
• Ratios of partition functions for the log-gamma polymer
• Full counting statistics for interacting trapped fermions
• Poisson--Dirichlet distribution for random Belyi surfaces
• One-dimensional stochastic growth and Gaussian ensembles of random matrices
• Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices
• Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles
• Breaking supersymmetry in a one-dimensional random Hamiltonian
• Tracy-Widom distribution as instanton sum of 2D IIA superstrings
• Painleve formulas of the limiting distributions for non-null complex sample covariance matrices
• Instantons and Extreme Value Statistics of Random Matrices
• A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation
• Discrete spacetime and its applications
• Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
• Increasing and Decreasing Subsequences of Permutations and Their Variants
• A fully noncommutative Painlevé II hierarchy: Lax pair and solutions related to Fredholm determinants
• Matrix Kernels for the Gaussian Orthogonal and Symplectic Ensembles
• Central limit theorem for linear spectral statistics of general separable sample covariance matrices with applications
• Exact Recovery of Community Detection in k-partite Graph Models
• The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants
• Variational formulas and cocycle solutions for directed polymer and percolation models
• The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs
• Finite n Largest Eigenvalue Probability Distribution Function of Gaussian Ensembles
• Hirota bilinear approach to GUE, NLS, and Painlevé IV
• Limiting Spectral Radii of Circular Unitary Matrices under Light Truncation
• Spectral Radii of Truncated Circular Unitary Matrices
• Spectral Radii of Products of Random Rectangular Matrices
• Disordered statistical physics in low dimensions: extremes, glass transition, and localization
• A universality property for last-passage percolation paths close to the axis
• Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble