A Stirling-type formula for the distribution of the length of longest increasing subsequences, applied to finite size corrections to the random matrix limit

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 tables of exact values (that have recently been compiled for $n$ up to $750$) 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) an expansion of the expected value of the length, exhibiting three more terms than previously known.