Asymptotic expansions relating to the distribution of the length of longest increasing subsequences

We study the distribution of the length of longest increasing subsequences in random permutations of n integers as n grows large and establish an asymptotic expansion in powers of $n^{-1/3}$ . Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy–Widom distribution...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2024-03, Vol.12, Article e36
Main Author: Bornemann, Folkmar
Format: Article
Language:English
Subjects:
05A16
30D15
30E15
33C10
60B20
Asymptotic series
Eigenvalues
Expected values
Growth models
Mathematical analysis
Permutations
Polynomials
Probability
Random variables
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Summary:We study the distribution of the length of longest increasing subsequences in random permutations of n integers as n grows large and establish an asymptotic expansion in powers of $n^{-1/3}$ . Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy–Widom distribution F, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of F with rational polynomial coefficients. Our proof replaces Johansson’s de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted to an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2024.13