Asymptotic expansions relating to the lengths of longest monotone subsequences of involutions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909914494926848 |
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| author | Bornemann, Folkmar |
| author_facet | Bornemann, Folkmar |
| contents | 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 $F_β$ 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 $F_β$ 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$β$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 $β=2$.) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_03798 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansions relating to the lengths of longest monotone subsequences of involutions Bornemann, Folkmar Probability Mathematical Physics Combinatorics 05A16, 60B20, 30D15, 30E15, 33C10 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 $F_β$ 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 $F_β$ 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$β$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 $β=2$.) |
| title | Asymptotic expansions relating to the lengths of longest monotone subsequences of involutions |
| topic | Probability Mathematical Physics Combinatorics 05A16, 60B20, 30D15, 30E15, 33C10 |
| url | https://arxiv.org/abs/2306.03798 |