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On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2018 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 31, no 4, p. 1155-1196Article in journal (Refereed) Published
Abstract [en]

We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the n x n Laguerre unitary ensemble. The probability that this ratio is greater than r > 1 is expressed in terms of an n x n Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as n -> infinity is found as an integral over (0, infinity) containing two functions q(1)(x) and q(2)(x). These functions satisfy a system of two coupled Painleve V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We compute asymptotic behaviours of these functions as rx -> 0(+) and (r - 1)x -> infinity, as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD , 2018. Vol. 31, no 4, p. 1155-1196
Keywords [en]
mathematical physics, random matrix theory, Riemann-Hilbert problems
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-224002DOI: 10.1088/1361-6544/aa9d57ISI: 000425666100002Scopus ID: 2-s2.0-85045029412OAI: oai:DiVA.org:kth-224002DiVA, id: diva2:1190543
Note

QC 20180314

Available from: 2018-03-14 Created: 2018-03-14 Last updated: 2018-03-14Bibliographically approved

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  • Other locale
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