The random-cluster model on the complete graph
type: Item
instance of: scholarly article
title: The random-cluster model on the complete graph
zbMATH Open document ID: 0846.05075
DOI: 10.1007/BF01213683
author: Béla Bollobás
author: G. R. Grimmett
author: Svante Janson
publication date: 1996-09-29T00:00:00Z
review text: The model under study in this paper is a refinement of the classical random graph \({\mathcal G}(n, p)\) of Erdös and Rényi [see \textit{B. Bollobás}, Random graphs (1985; Zbl 0567.05042)]. Recall that in this model of random graph on a set \(V\) of \(n\) vertices, the events \(A_{ij}= \{\{i, j\}\mid \{i, j\}\) is an edge of the graph\}, where \(\{i, j\}\) runs over all two-elements subsets of \(V\), are all independent, and have probability \(p\). This model is here modified by introducing a new parameter \(q\), which is a positive real, and the probability distribution of the random graph \({\mathcal G}(n, p, q)\) is obtained from \({\mathcal G}(n, p)\) by putting on each graph a weight proportional to \(q^c\), where \(c\) is the number of connected components of the graph. As is explained in the introduction, the main motivation for studying such a random graph model is that this is the mean-field version of the random-cluster model introduced by C. M. Fortuin and P. W. Kasteleyn around 1970. The main results on the random graph model \({\mathcal G}(n, p, q)\) concern the behaviour of the largest component of the graph \({\mathcal G}(n, {\lambda\over n}, q)\), for \(\lambda> 0\), when \(n\to \infty\). In the Erdös-Rényi model (i.e. for \(q= 1\)), one knows that the value \(\lambda= 1\) is critical for the appearance of a giant component (i.e. a component with a number of vertices of the order \(n\)). In the \({\mathcal G}(n, {\lambda\over n}, q)\) model, there is for each value of \(q\) an explicit critical value of \(\lambda\), such that a giant component appears for \(\lambda> \lambda_c(q)\). More precisely, the following results are shown: (a) for \(0< \lambda< \lambda_c(q)\), the largest component has order \(\log n\); (b) for \(\lambda> \lambda_c(q)\), the largest component has order \(\theta n\), while the other components have order smaller than \(\log n\), where \(\theta\) is an explicit function of \(\lambda\) and \(q\); (c) in the critical case \(\lambd…
Mathematics Subject Classification ID: 05C80
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 878008
zbMATH Keywords: complete graph
zbMATH Keywords: random graph
zbMATH Keywords: probability distribution
zbMATH Keywords: random-cluster model
zbMATH Keywords: giant component
zbMATH Keywords: colouring
reviewed by: Philippe Biane
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 878008