Limit shapes for Gibbs ensembles of partitions

TL;DR

This paper explicitly computes the limit shapes for various Gibbs ensembles of integer partitions, revealing that all possible shapes fall into distinct classes based on the asymptotics of internal energies, with applications in aggregation models.

Contribution

It provides explicit formulas for limit shapes of Gibbs ensembles of partitions and classifies them based on internal energy asymptotics, connecting to aggregation and coagulation models.

Findings

All limit shapes fall into several distinct classes.

Limit shapes are determined by the asymptotics of internal energies.

Connections to invariant measures of zero range and coagulation-fragmentation processes.

Abstract

We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation processes. We show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.