Unlabelled Gibbs partitions

TL;DR

This paper investigates unlabelled Gibbs partitions, focusing on the emergence of a giant component and the distribution of small fragments, providing a probabilistic framework for weighted combinatorial structures.

Contribution

It introduces a general model for unlabelled Gibbs partitions, analyzing the formation of a giant component and the asymptotic distribution of small fragments.

Findings

Existence of a giant component in the model

Small fragments converge to a Pólya-Boltzmann distribution

Framework encompasses multisets of weighted objects

Abstract

We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of weighted combinatorial objects. We describe a general setting characterized by the formation of a giant component. The collection of small fragments is shown toconverge in total variation toward a limit object following a P\'olya-Boltzmann distribution.