2023/06/06 by Folkmar Bornemann, Bornemann, Folkmar · 1 citation
Computer Science · Mathematics · #05A16 #30D15 #30E15 #33C10 #60B20 #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Distribution Estimation and Applications
paper · pdf · doi:10.48550/arxiv.2306.03798
openalex publication_date 2023/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Fβ for β=1 or β=4, we find explicit analytic expressions of the first few expansion terms as linear combinations of higher order derivatives of Fβ 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 LβE 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 β=2.)