Gibbs partitions: the convergent case

TL;DR

This paper analyzes Gibbs partitions with a dominant giant component, showing convergence of small fragments to Boltzmann-distributed structures, and applies these results to graph classes, extending previous work on random graphs.

Contribution

It introduces a unified framework for Gibbs partitions with a giant component and applies it to small block-stable graph classes, generalizing prior results.

Findings

Gibbs partitions typically form a unique giant component.

Small fragments converge to Poisson Boltzmann limit graphs.

Results extend to proper addable minor-closed graph classes.

Abstract

We study Gibbs partitions that typically form a unique giant component. The remainder is shown to converge in total variation toward a Boltzmann-distributed limit structure. We demon- strate how this setting encompasses arbitrary weighted assemblies of tree-like combinatorial structures. As an application, we establish smooth growth along lattices for small block-stable classes of graphs. Random graphs with n vertices from such classes are shown to form a giant connected component. The small fragments may converge toward different Poisson Boltzmann limit graphs, depending along which lattice we let n tend to infinity. Since proper addable minor-closed classes of graphs belong to the more general family of small block-stable classes, this recovers and generalizes results by McDiarmid (2009).