Asymptotic expansions relating to the lengths of longest monotone subsequences of involutions
| Autor: | Bornemann, Folkmar |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1080/10586458.2024.2397334 |
| Popis: | 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 $\beta=2$.) Comment: 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 |
| Databáze: | arXiv |
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