Universality of the distribution functions of random matrix theory

Abstract

In probability theory and statistics a common first approximation to many random processes is a sequence X 1 , X 2 , X 3 , . . . of independent and identically distributed (iid) random variables. Let F denote their common distribution. To motivate the material below, we take these random variables and construct a particularly simpleThe order statistics are the eigenvalues orderedand the distribution of the largest eigenvalue, λ max (N ) = λ N , is Prob (λ max (N ) ≤ x) = Prob (X 1 ≤ x, . . . , X N ≤ x) = F (x) N .

An Introduction to Random Matrices

“…These involve the study of other types of orthogonal polynomials than the Hermite polynomials (e.g., Laguerre or Jacobi). See [For93], [For94], [TrW94b], [TrW00], [Joh00], [John01], [For06], and the book [For05].…”

Matrix models for beta ensembles

“…The edge scaling limit refers to the distribution of the largest eigenvalue of a matrix in the ensemble; the bulk scaling limit refers to the distribution of an eigenvalue in the "bulk" of the spectrum. For a reference, see [29], [30] or [28]. The Tracy-Widom distributions are defined in terms of Painlevé functions, which are solutions to certain differential equations, with asymptotics given by Airy functions.…”

Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture

“…See [4], [43], [51]. Many of the results stated here for unitary matrices are "universal", applying to many other matrix ensembles (just as the central limit theorem): see Tracy-Widom [93] for an overview of the many great results. I have focused on eigenvalues of unitary or Hermitian matrices.…”

“…The work of Baik-Deift-Johansson on an integrable systems approach to largest eigenvalues and the longest increasing subsequence of a random permutation is surveyed in [27]. The work of Tracy-Widom on a wide variety of random matrix results and applications using Painlevé transcendents is surveyed in [90], [91] and [92], [93]. The work of Adler-Van Moerbeke linking random matrices to Virasoro algebras and much else is surveyed in [3].…”