[Paper Review] Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions

Cyril Banderier, Markus Kuba+1 more|arXiv (Cornell University)|Mar 5, 2021

Material Science and Thermodynamics4 citations

TL;DR

This paper introduces an extended critical composition scheme $ F(z,u) = M(z)G(uH(z)) $ to analyze phase transitions in combinatorial structures, revealing a universal three-parameter limit law—the beta-Mittag-Leffler distribution—through singularity analysis of generating functions. It establishes double phase transitions involving Mittag-Leffler, Boltzmann, and mixed Poisson laws, with unified threshold explanations and moment convergence results.

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 $F(z,u)=G(u H(z))M(z)$. 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.

Motivation & Objective

• To generalize the critical composition scheme $ F(z) = G(H(z)) $ to include a size-marking parameter $ u $ and an auxiliary function $ M(z) $, enabling analysis of component counts in combinatorial structures.
• To identify universal limit laws—particularly the beta-Mittag-Leffler distribution—emerging in critical composition schemes with singular exponents.
• To unify the understanding of phase transitions involving Mittag-Leffler, Boltzmann, and mixed Poisson laws through a common analytical framework.
• To extend the analysis to size-refined schemes and multivariate settings, capturing discrete-to-continuous phase transitions.
• To provide moment convergence and local limit theorems for the derived limit laws, ensuring probabilistic robustness.

Proposed method

• Uses singularity analysis of generating functions with Puiseux expansions to characterize asymptotics near dominant singularities.
• Applies the extended composition scheme $ F(z,u) = M(z)G(uH(z)) $, where $ u $ marks the number of $ H $-components, to model component counts in combinatorial families.
• Derives limit laws via asymptotic expansions of $ G $ and $ H $, particularly when both functions are simultaneously singular (critical case).
• Introduces the beta-Mittag-Leffler distribution as a product of beta and Mittag-Leffler laws, emerging as the universal limit in extended schemes.
• Applies the method to size-refined schemes by modifying $ H(z) $ to track components of a specific size, leading to mixed Poisson-type phase transitions.
• Extends results to multivariate and cyclic schemes, including $ F(z,u) = -\log(1 - uH(z)) $, yielding Mittag-Leffler limits.

Experimental results

Research questions

• RQ1 What universal limit laws emerge in critical composition schemes when the number of components is marked via a generating function parameter $ u $?
• RQ2 How do phase transitions in component counts arise from the interplay of singular exponents in $ G $ and $ H $, and what determines the thresholds?
• RQ3 What is the role of the beta-Mittag-Leffler distribution in unifying limit laws across different combinatorial models?
• RQ4 How do moment convergence and local limit theorems behave in these critical composition schemes?
• RQ5 Can the framework be extended to multivariate and cyclic schemes to capture richer phase transition phenomena?

Key findings

• The extended composition scheme $ F(z,u) = M(z)G(uH(z)) $ leads to the beta-Mittag-Leffler distribution as the universal limit law for component counts in critical combinatorial structures.
• Double phase transitions are identified, involving transitions between Mittag-Leffler, Boltzmann, and mixed Poisson laws, with thresholds determined by the singular exponents of $ G $ and $ H $.
• The size-refined scheme $ F(z,v) = M(z)G(H(z) - z^j h_j(1-v)) $ induces a mixed Poisson-type phase transition, capturing component size distributions.
• Moment convergence and local limit theorems are established for all derived limit laws, ensuring strong probabilistic convergence.
• The framework applies to diverse models including random walks, increasing trees, Pólya urns, and the Chinese restaurant process, with explicit limit laws derived for each.
• The multivariate extension $ F(z,\mathbf{u}) = M(z)\prod_{\ell=1}^m G_\ell(u_\ell H_\ell(z)) $ yields multivariate product distributions, generalizing the univariate results.