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Gibbs partitions: the convergent case

Benedikt Stufler

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 demonstrate 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).
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Dates and versions

ensl-01408153 , version 1 (03-12-2016)

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Benedikt Stufler. Gibbs partitions: the convergent case. 2016. ⟨ensl-01408153⟩

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