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Random-cluster measures and uniform spanning trees. (English) Zbl 0840.60089
C. M. Fortuin and P. W. Kasteleyn [Phys. 57, 536-564 (1972)] introduced the random cluster model which is a two-parameter family of processes on the integer lattice \(Z^d\), including independent bond percolation and Potts models. In the present paper it is shown that the random-cluster model in a certain sense also includes the uniform spanning tree measure which was introduced by R. Pemantle [Ann. Probab. 19, No. 4, 1559-1574 (1991; Zbl 0758.60010)]. Consider a fixed undirected graph \(G\) with finite vertex set \(V\) and edge set \(E\). Any subgraph of \(G\) is then uniquely determined by its edge set which can be identified with an element of \(\{0,1\}^E\). Now let \(0 \leq p \leq 1\) and \(q > 0\) be two parameters. The random cluster measure \(\mu^{p, q}_G\) is a probability measure on the power set of \(\{0,1\}^E\) assigning any subgraph \(\eta\) of \(G\) the probability \(\mu_G^{p,q} (\eta)\) proportional to \(\{\prod_{e \in E} (p^{\eta (e)} (1 - p)^{1 - \eta (e)})\} q^{k (\eta)}\), \(k (\eta)\) denoting the number of connected components of \(\eta\). The author shows that if \(p \to 0\), \(q \to 0\) such that \(q/p \to 0\), then \(\mu^{p,q}_G\) converges weakly to the uniform spanning tree measure \(\mu^U_G\) concentrated on the power set of spanning trees for \(G\). Finally it is shown that an analogous result holds for the graph \(Z^d\), the edges being the pairs of nearest neighbours.
Reviewer: K.Schürger (Bonn)

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60B10 Convergence of probability measures
60C05 Combinatorial probability
Full Text: DOI
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