On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble

We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the Laguerre unitary ensemble. The probability that this ratio is greater than r  >  1 is expressed in terms of an Hankel determinant with a perturbed Laguerre weight. The limiting probabilit...

Full description

Saved in:
Bibliographic Details
Published in:Nonlinearity 2018-04, Vol.31 (4), p.1155-1196
Main Authors: Atkin, Max R, Charlier, Christophe, Zohren, Stefan
Format: Article
Language:English
Subjects:
mathematical physics
random matrix theory
Riemann-Hilbert problems
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the Laguerre unitary ensemble. The probability that this ratio is greater than r  >  1 is expressed in terms of an Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as is found as an integral over containing two functions q1(x) and q2(x). These functions satisfy a system of two coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We compute asymptotic behaviours of these functions as and , as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x.
ISSN:0951-7715
1361-6544
1361-6544
DOI:10.1088/1361-6544/aa9d57