Gibbs partitions and lattice paths

TL;DR

This paper analyzes a Gibbs partition model with a new component weight condition, revealing a condensation phenomenon with a giant component and a power-law distribution of smaller components, applicable to various combinatorial models.

Contribution

It introduces a novel condition on component weights in Gibbs partitions, leading to new asymptotic behaviors and a limit theorem for component sizes.

Findings

Discovery of a condensation phenomenon with a giant component

Asymptotic power-law growth of non-maximal components

Application to diverse combinatorial models such as lattice paths and urns

Abstract

This work is devoted to the analysis of a Gibbs partition model, also known as a composition scheme. We consider a natural new condition on the component weights. It leads to a new behavior for the total number of components. We discover a condensation phenomenon, producing a unique giant component comprising almost the entire mass. Additionally, we prove a point process limit describing the asymptotic size of the non-maximal components exhibiting a sublinear power-law growth. A particular motivation for our article stems from applications, ranging from simple random walks in the cube, over lattice paths models in the plane, pairs of directed random walks, over to urn models and card guessing games.