Uploaded November 2025 | Updated September 2026, 3 weeks ago
Composition schemes are ubiquitous in combinatorics, number theory, 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 (models 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 ($q\gtq_c$), and it is a Boltzmann distribution in the subcritical regime ($0\ltq\ltq_c$). 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.
(Based on the article arxiv.org/abs/2311.17226 by Cyril Banderier, Markus Kuba, Stephan Wagner, Michael Wallner).
Cyril Banderier (LIPN)
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Find this and many more scientific videos on carmin.tv - a French video platform for mathematics and their interactions with other sciences offering extra functionalities tailored to meet the needs of the research community.
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Composition schemes are ubiquitous in combinatorics, number theory, 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 (models 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 ($q\gtq_c$), and it is a Boltzmann distribution in the subcritical regime ($0\ltq\ltq_c$). 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.
(Based on the article arxiv.org/abs/2311.17226 by Cyril Banderier, Markus Kuba, Stephan Wagner, Michael Wallner).
Cyril Banderier (LIPN)
===
Find this and many more scientific videos on carmin.tv - a French video platform for mathematics and their interactions with other sciences offering extra functionalities tailored to meet the needs of the research community.
===
