Limit shapes for Gibbs partitions of sets

TL;DR

This paper investigates the asymptotic limit shapes of partitions of sets within Gibbs ensembles, extending previous integer partition results to set partitions involving sums of Poisson variables, and analyzes various energy behavior scenarios.

Contribution

It introduces the computation of limit shapes for set partitions in Gibbs ensembles, broadening the understanding from integer partitions to set partitions with Poisson sums.

Findings

Derived limit shapes for set partitions under Gibbs measures

Analyzed scenarios with different energy asymptotics

Computed local limit shape profiles for step functions

Abstract

This study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand canonical Gibbs ensembles of partitions of sets, which lead to the sums of Poisson random variables. Under mild monotonicity assumptions, we study all possible scenarios arising from different asymptotic behaviors of the energy, and also compute local limit shape profiles for cases in which the limit shape is a step function.