Gibbs partitions: The convergent case. Issue 3 (27th February 2018)

• Record Type: Journal Article
• Authors: Stufler, Benedikt
• Is Part Of: Random structures & algorithms. Volume 53:Issue 3(2018)
• Page Start: 537
• Page End: 558
• Publication Date: 2018-02-27
• Subjects: Gibbs partitions -- graph classes -- graph limits -- random graphs -- random partitions of sets; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20771
• Languages: English
• ISSNs: 1042-9832

Abstract

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).