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.
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Shinault, G., Tracy, C.A. Asymptotics for the Covariance of the Airy2 Process. J Stat Phys 143, 60–71 (2011). https://doi.org/10.1007/s10955-011-0155-5
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DOI: https://doi.org/10.1007/s10955-011-0155-5