Asymptotic expansions relating to the lengths of longest monotone subsequences of involutions

Abstract

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 expansion 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 arXiv:2301.02022, which established similar results in the case of general permutations and $\beta =2$.)

V6: updated discussion of the hard-to-soft edge limit with a thinning parameter; Julia code for large scale Monte-Carlo simulations included; 50 pages, 5 figures, 3 tables

References in corpus (3)

• A determinantal formula for the GOE Tracy-Widom distribution
• A Stirling-type formula for the distribution of the length of longest increasing subsequences
• Asymptotic expansions relating to the distribution of the length of longest increasing subsequences

Cited by in corpus (1)

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