Asymptotics for the Covariance of the Airy_2 process

Details Asymptotics for the Covariance of the Airy_2 process Asymptotics for the Covariance of the Airy_2 process

2010-11-30

arxiv.org/abs/1011.6616v2

J. Statistical Physics 143 (2011), 60-71

Mathematics

Probability

14 pages. Version 2 adds some clarifying remarks and updates the references

Scientific paper

10.1007/s10955-011-0155-5

In this paper we compute some of the higher order terms in the large-t asymptotic expansion of the Airy process two-point function, extending the previous work of Adler and van Moerbeke and Widom. We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Hastings-McLeod Painlev\'e II function and its first derivative. 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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