Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions

Abstract

Multitudinous probabilistic and combinatorial objects are associated with generating functions satisfying a composition scheme $F(z)=G(H(z))$. The analysis becomes challenging when this scheme is critical (i.e., $G$ and $H$ are simultaneously singular). Motivated by many examples (random mappings, planar maps, directed lattice paths), we consider a natural extension of this scheme, namely $F(z,u)=G(uH(z))M(z)$. We also consider a variant of this scheme, which allows us to analyse the number of $H$-components of a given size in $F$. We prove that these two models lead to a rich world of limit laws, where we identify the key role played by a new universal law introduced in this article: the three-parameter Mittag-Leffler distribution, which is essentially the product of a beta and a Mittag-Leffler distribution. We also prove (double) phase transitions, additionally involving Boltzmann and mixed Poisson distributions, bringing a unified explanation of the associated thresholds. In all cases we obtain moment convergence and local limit theorems. We end with extensions of the critical composition scheme to a cycle scheme and to the multivariate case, leading to product distributions. Applications are presented for random walks, trees (supertrees of trees, increasingly labelled trees, preferential attachment trees), triangular Pólya urns, and the Chinese restaurant process.

59 pages; dedicated to Alois Panholzer on the occasion of his 50th birthday. Published in the Annals of Applied Probability at https://doi.org/10.1214/24-AAP2076

References in corpus (18)

• The Prabhakar or three parameter Mittag--Leffler function: theory and application
• Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics
• Analytic urns
• Lamperti-type laws
• Degree asymptotics with rates for preferential attachment random graphs
• Singularity analysis, Hadamard products, and tree recurrences
• On the complete monotonicity of the three parameter generalized Mittag-Leffler function $E_{\alpha ,\beta }^{\gamma }(-x)$
• Generalized gamma approximation with rates for urns, walks and trees
• Enumeration and asymptotics of restricted compositions having the same number of parts
• A half-normal distribution scheme for generating functions
• Generalized Mittag Leffler distributions arising as limits in preferential attachment models
• A Berry-Esseen theorem for Pitman's $\alpha$-diversity
• Local time for lattice paths and the associated limit laws
• Periodic Pólya Urns, the Density Method, and Asymptotics of Young Tableaux
• Boltzmann sampling of irreducible context-free structures in linear time
• Some reflections on directed lattice paths
• Composition schemes: q-enumerations and phase transitions
• Convolutions and Mixtures of Gamma, Stable and Mittag-Leffler Distributions