Journal of Statistical Physics

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

Asymptotics for the Covariance of the Airy2 Process

Open Access


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.


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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