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Different Flavors of Asymptotics in Random Permutations and Their Impact on Computing Finite-Size Effects

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ARA2 - Applicable resurgent asymptotics: towards a universal theory

The distribution of the length of longest increasing subsequences in random permutations/involutions of the symmetric group S_n is positioned in a rich web of knowledge connecting, e.g., constructive combinatorics, random matrix theory, integrals over classical groups, Toeplitz/Hankel determinants, Riemann-Hilbert problems, Painlevé/Chazy equations, and operator determinants. The known techniques for establishing a meaningful double-scaling limit near the mode of the length distribution (in terms of the Tracy-Widom distributions for beta=1, 2, 4) use a Tauberian argument, called de-Poissonization, which does not render itself to establish asymptotic expansions. Recently Forrester and Mays have started studying the structure of finite-size effects numerically and visualized the coarse form of the first such term based on data from Monte-Carlo simulations for n up to 105. In this talk we show that the theory of Hayman admissibility yields a different, less explicit but numerically highly accessible asymptotics that gives blazingly fast, surprisingly robust and accurate results — outperforming combinatorial methods and the random matrix asymptotics in the mesoscopic regime (for, say, n up to 10{12}). It allows to approximate the first two finite-size corrections to the random matrix limit. We derive, heuristically, expansions of the expected value and variance of the length distribution, exhibiting several more terms than previously known.

This talk is part of the Isaac Newton Institute Seminar Series series.

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