Asymptotics of the largest eigenvalue distribution of the Laguerre unitary ensemble

2020/01/31 by Shulin Lyu, Chao Min, Yang Chen
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Combinatorics #Distribution (mathematics) #Eigenvalues and eigenvectors #Hermite polynomials #Hermitian matrix #Interval (graph theory) #Laguerre polynomials #Mathematical analysis #Mathematical functions and polynomials #Mathematical physics #Mathematics #Operator (biology) #Orthogonal polynomials #Physics #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Unitary matrix #Unitary state #math-ph #math.MP #msc:15B52 #msc:41A60 #msc:42C05

paper · pdf · doi:10.1063/5.0010029

18 pages

arxiv created 2020/03/29 · openalex publication_date 2021/06/01 · arxiv updated 2021/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight xγe−4nx,x∈0,∞,γ>−1, lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distribution function with α ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159, 151–174 (1994)] and later proved by Deift, Its, and Krasovsky [Commun. Math. Phys. 278, 643–678 (2008)]. Our conclusions are reduced to those of Deift et al. when γ = 0. It should be pointed out that our derivation is straightforward but not rigorous, and hence, the above results are stated as conjectures.

Citations

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