# zbMATH — the first resource for mathematics

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)

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60B10 Convergence of probability measures 60C05 Combinatorial probability
##### Keywords:
bond percolation; Potts models; random cluster measure
Full Text:
##### References:
 [1] Aizenman, M.; Chayes, J.T.; Chayes, L.; Newman, C.M., Discontinuity of the magnetization in one-dimensional 1/|x − y|2 Ising and Potts models, J. statist. phys., 50, 1-40, (1988), 1988 · Zbl 1084.82514 [2] Aldous, D., The random walk construction of uniform spanning trees and uniform labeled trees, SIAM J. discrete. math., 3, 450-465, (1990) · Zbl 0717.05028 [3] Burton, R.; Keane, M., Density and uniqueness in percolation, Comm. math. phys., 121, 501-505, (1989) · Zbl 0662.60113 [4] Burton, R.; Pemantle, R., Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer impedances, Ann. probab., 21, 1329-1371, (1993) · Zbl 0785.60007 [5] Doyle, P.; Snell, J.L., Random walks and electric networks, () · Zbl 0583.60065 [6] Fortuin, C.M., A on the random-cluster model. II, The percolation model phys., 58, 393-418, (1972) [7] Fortuin, C.M., B on the random-cluster model. III, The simple random-cluster process, phys., 59, 545-570, (1972) [8] Fortuin, C.M.; Kasteleyn, P.W., On the random-cluster model. I. introduction and relation to other models, Phys., 57, 536-564, (1972) [9] Georgii, H., Gibbs measures and phase transitions, (1988), de Gruyter New York · Zbl 0657.60122 [10] Grimmett, G., Percolation, (1989), Springer New York · Zbl 0691.60089 [11] Grimmett, G., The stochastic random-cluster process, and the uniqueness of random-cluster measures, (1994), to appear in Ann. Probab · Zbl 0858.60093 [12] Grimmett, G., Comparison and disjoint-occurrence inequalities for random-cluster models, J. statist. phys., 78, 1311-1324, (1995) · Zbl 1080.82546 [13] Pemantle, R., Choosing a spanning tree for the integer lattice uniformly, Ann. probab., 19, 1559-1574, (1991) · Zbl 0758.60010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.