Asymptotics of the Airy-kernel determinant
type: Item
instance of: scholarly article
title: Asymptotics of the Airy-kernel determinant
zbMATH Open document ID: 1167.15005
published in: Communications in Mathematical Physics
publication date: 2008-11-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0609451
review text: Let \(K_s\) be the trace-class operator with kernel \[ K_s(t,u)=\frac{\text{Ai}(t)\text{Ai}'(u)-\text{Ai}(u)\text{Ai}'(t)}{t-u} \] acting on \(L^2(-s,\infty)\), where \(\text{Ai}(x)\) is the Airy function. The paper is concerned with the behavior of the Fredholm determinant \(\text{det}(I-K_s)\) as \(s\rightarrow +\infty\). In fact, the authors prove that the large \(s\)-asymptotic behavior of this determinant is given by \[ \ln \text{det}(I-K_s)=-\frac{s^3}{12}-\frac{1}{8}\ln s+\chi +O(s^{-3/2}), \] where \[ \chi=\frac{1}{24}\ln 2+\zeta'(-1) \] and \(\zeta(s)\) is the Riemann zeta-function. The value of the constant \(\chi\) is computed using a Riemann-Hilbert approach.
reviewed by: Isabel Marrero
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 82B05
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 5373466
zbMATH Keywords: Random matrix
zbMATH Keywords: Airy function
zbMATH Keywords: Airy-kernel determinant
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: trace-class operator
zbMATH Keywords: Fredholm determinant
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: Riemann zeta-function
author: Percy A. Deift
author: A. R. Its
author: I. V. Krasovsky
MaRDI profile type: Publication
OpenAlex ID: W2094621289
arXiv ID: math/0609451
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DOI: 10.1007/S00220-007-0409-X
description: scientific article