On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble. (19th February 2018) - Atkin, Max R; Charlier, Christophe; Zohren, Stefan
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 .