Limit shapes of multiplicative measures associated with coagulation-fragmentation processes and random combinatorial structures

TL;DR

This paper establishes limit shapes for multiplicative measures on partitions linked to coagulation-fragmentation processes and combinatorial assemblies, analyzing fluctuations and phase transitions in component independence.

Contribution

It introduces new limit shape results for measures induced by exponential generating functions with expansive parameters, connecting coagulation-fragmentation dynamics and combinatorial structures.

Findings

Proves the functional central limit theorem for scaled partition fluctuations.

Identifies a threshold where component independence becomes conditional.

Discusses the relationship between limit shapes, thresholds, and gelation phenomena.

Abstract

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, $a_k\sim Ck^{p-1}, k\to\infty, p>0$,where $C$ is a positive constant. The measures considered are associated with reversible coagulation-fragmentation processes and certain combinatorial structures, known as assemblies. We prove the functional central limit theorem for the fluctuations of a scaled random partition from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence. Among other things, the paper also discusses, in a general setting, the interplay between limit shapes, threshold and gelation.