Composition schemes: q-enumerations and phase transitions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913437565583360 |
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| author | Banderier, Cyril Kuba, Markus Wagner, Stephan Wallner, Michael |
| author_facet | Banderier, Cyril Kuba, Markus Wagner, Stephan Wallner, Michael |
| contents | Composition schemes are ubiquitous in combinatorics, statistical mechanics and probability theory. We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of~$q$-enumeration (this is a model where objects with a parameter of value $k$ have a Gibbs measure/Boltzmann weight $q^k$). For structures enumerated by a composition scheme, we prove a phase transition for any parameter having such a Gibbs measure: for a critical value $q=q_c$, the limit law of the parameter is a two-parameter Mittag-Leffler distribution, while it is Gaussian in the supercritical regime ($q>q_c$), and it is a Boltzmann distribution in the subcritical regime ($0<q<q_c$). We apply our results to fundamental statistics of lattice paths and quarter-plane walks. We also explain previously observed limit laws for pattern-restricted permutations, and a phenomenon uncovered by Krattenthaler for the wall contacts in watermelons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_17226 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Composition schemes: q-enumerations and phase transitions Banderier, Cyril Kuba, Markus Wagner, Stephan Wallner, Michael Combinatorics Probability 60C05, 60E07, 60G50 Composition schemes are ubiquitous in combinatorics, statistical mechanics and probability theory. We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of~$q$-enumeration (this is a model where objects with a parameter of value $k$ have a Gibbs measure/Boltzmann weight $q^k$). For structures enumerated by a composition scheme, we prove a phase transition for any parameter having such a Gibbs measure: for a critical value $q=q_c$, the limit law of the parameter is a two-parameter Mittag-Leffler distribution, while it is Gaussian in the supercritical regime ($q>q_c$), and it is a Boltzmann distribution in the subcritical regime ($0<q<q_c$). We apply our results to fundamental statistics of lattice paths and quarter-plane walks. We also explain previously observed limit laws for pattern-restricted permutations, and a phenomenon uncovered by Krattenthaler for the wall contacts in watermelons. |
| title | Composition schemes: q-enumerations and phase transitions |
| topic | Combinatorics Probability 60C05, 60E07, 60G50 |
| url | https://arxiv.org/abs/2311.17226 |