On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble. (19th February 2018)

• Record Type: Journal Article
• Authors: Atkin, Max R; Charlier, Christophe; Zohren, Stefan
• Is Part Of: Nonlinearity. Volume 31:Number 4(2018:Apr.)
• Page Start: 1155
• Page End: 1196
• Publication Date: 2018-02-19
• Subjects: mathematical physics -- random matrix theory -- Riemann–Hilbert problems; 60B20 -- 15B52 -- 42C05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa9d57
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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 q 1 ( x ) and q 2 ( 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 .