The random-cluster model on the complete graph
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 \(\lambda= \lambda_c(q)\), two subcases appear, first if \(0< q\leq 2\), then the largest component has order \(n^{2/3}\), while for \(q> 2\), the graph is either as in the subcritical or in the uppercritical case. Here the statements mean that the events alluded to occur are of a probability which goes to one as \(n\to \infty\). The proof of these statements rely on a colouring lemma, which, roughly, asserts that the random graph \({\mathcal G}(n, p, q)\) can be obtained from an Erdös-Rényi random graph, by two-colouring randomly the components of the graph, and the conditioning on the number of edges of one colour to be \(n\). Some other quantities of interest are also evaluated, such as the ``free energy'', or the density of edges, and some applications are made to large deviation results for Erdös-Rényi random graphs.
scientific article; zbMATH DE number 878008
scientific article