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Journal of Statistical Physics

, Volume 143, Issue 1, pp 60–71 | Cite as

Asymptotics for the Covariance of the Airy2 Process

  • Gregory Shinault
  • Craig A. Tracy
Open Access
Article

Abstract

In this paper we compute some of the higher order terms in the asymptotic behavior of the two point function \(\mathbb{P}(\mathcal {A}_{2}(0)\leq s_{1},\mathcal{A}_{2}(t)\leq s_{2})\), extending the previous work of Adler and van Moerbeke (arXiv:math.PR/0302329; Ann. Probab. 33, 1326–1361, 2005) and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Painlevé II function q and its derivative q′. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the Tracy-Widom GUE density function f 2 and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to tenth order in terms of the moments of the Tracy-Widom GUE distribution.

Keywords

Airy process Asymptotics 

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA

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