Asymptotic Expansions Relating to the Lengths of Longest Monotone Subsequences of Involutions

We study the distribution of the length of longest monotone subsequences in random (fixed-point free) involutions of n integers as n grows large, establishing asymptotic expansions in powers of n-1/6 in the general case and in powers of n-1/3 in the fixed-point free cases. Whilst the limit laws were shown by Baik and Rains to be one of the Tracy-Widom distributions F beta for beta = 1 or beta = 4, we find explicit analytic expressions of the first few expansions terms as linear combinations of higher order derivatives of F beta with rational polynomial coefficients. Our derivation is based on a concept of generalized analytic de-Poissonization and is subject to the validity of certain hypotheses for which we provide compelling (computational) evidence. In a preparatory step expansions of the hard-to-soft edge transition laws of L beta E are studied, which are lifted into expansions of the generalized Poissonized length distributions for large intensities. (This paper continues our work [13], which established similar results in the case of general permutations and beta = 2.)