Appears in collection : Combinatorics and Arithmetic for Physics

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 ($q>q_c$), 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. (Based on the article https://arxiv.org/abs/2311.17226 by Cyril Banderier, Markus Kuba, Stephan Wagner, Michael Wallner).