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Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models

Authors Cyril Banderier , Markus Kuba , Stephan Wagner , Michael Wallner

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Author Details

Cyril Banderier
  • Laboratoire d'Informatique de Paris Nord, Université Sorbonne Paris Nord, Villetaneuse, France
Markus Kuba
  • Department Applied Mathematics & Physics, University of Applied Sciences - Technikum Wien, Austria
Stephan Wagner
  • Institute of Discrete Mathematics, TU Graz, Austria
  • Department of Mathematics, Uppsala Universitet, Sweden
Michael Wallner
  • Institut für Diskrete Mathematik und Geometrie, TU Wien, Austria

Funding

  • Banderier, Cyril: supported by the French-Austrian PHC Amadeus project "Asymptotic behaviour of combinatorial structures".
  • Wagner, Stephan: supported by the Swedish research council (VR), grant 2022-04030.
  • Wallner, Michael: supported by the Austrian Science Fund (FWF): P 34142 and OeAD WTZ project FR 01/2023.

Acknowledgements

The second author warmly thanks Thomas Feierl for many discussions about watermelons and Paul Schreivogl for discussions on partition functions. We also thank Christian Krattenthaler for drawing our attention to the open problem of the phase transition of contacts in watermelons, and the two referees for their feedback. Last but not least, we are pleased to dedicate this excursion into the realm of composition schemes and limit laws to one of the greatest experts of this realm, namely Michael Drmota, at the occasion of his 60th birthday!

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Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner. Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.7

Abstract

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Distribution functions
Keywords
  • Composition schemes
  • q-enumeration
  • generating functions
  • Gibbs distribution
  • phase transitions
  • limit laws
  • Mittag-Leffler distribution
  • chi distribution
  • Boltzmann distribution

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