2026-07-24T09:41:09Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20442 2024-07-18T12:12:42Z ddc:004 open_access

Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models Banderier, Cyril Kuba, Markus Wagner, Stephan Wallner, Michael Composition schemes q-enumeration generating functions Gibbs distribution phase transitions limit laws Mittag-Leffler distribution chi distribution Boltzmann distribution 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Cyril Banderier and Markus Kuba and Stephan Wagner and Michael Wallner 2024 Is Part Of LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.AofA.2024.7 urn:nbn:de:0030-drops-204427 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.7 eng https://creativecommons.org/licenses/by/4.0/legalcode