A Stirling-type formula for the distribution of the length of longest increasing subsequences, applied to finite size corrections to the random matrix limit
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The discrete distribution of the length of longest increasing subsequences in random permutations of order n is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of the distribution of the largest level of the large matrix limit of GUE. As a numerical approximation, however, this asymptotics is inaccurate for small lengths and has a slow convergence rate, conjectured to be just of order n^-1/3. Here, we suggest a different type of approximation, based on Hayman's generalization of Stirling's formula. Such a formula gives already a couple of correct digits of the length distribution for n as small as 20 but allows numerical evaluations, with a uniform error of apparent order n^-3/4, for n as large as 10^12; thus closing the gap between a table of exact values (that has recently been compiled for up to n=1000) and the random matrix limit. Being much more efficient and accurate than Monte-Carlo simulations for larger n, the Stirling-type formula allows for a precise numerical understanding of the first few finite size correction terms to the random matrix limit, a study that has recently been initiated by Forrester and Mays, who visualized the form of the first such term. We display also the second one, of order n^-2/3, and derive (heuristically) expansions of expected value and variance of the length, exhibiting several more terms than previously known.
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