The Random Cluster Model is a mathematical model used primarily in statistical physics and probability theory to study statistical properties of systems exhibiting phase transitions. It is particularly relevant for understanding percolation, critical phenomena, and other related concepts in network theory and social dynamics. ### Basic Concepts: 1. **Clusters**: In the context of the model, a "cluster" refers to a group of connected nodes or sites in a network or lattice.
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On a finite graph , the random-cluster model weights an open-edge configuration byThe partition function is the sum of these weights, and counts the open connected components of a graph, including isolated graph vertices. The endpoint laws at and are obtained by limits. The factor is the cluster weight in the random-cluster model.[ref]
On the random-cluster model. I. Introduction and relation to other models by C. M. Fortuin and P. W. Kasteleyn. Finite-graph random-cluster measure and relation to other lattice models
At , the component factor disappears and the law is independent bond percolation. For other , is a model parameter rather than necessarily the marginal opening probability of each edge. For a single edge joining two vertices, with , the closed and open weights are and , so the edge is open with probability .
The random-cluster single-edge conditional probability describes the local mechanism. The Edwards-Sokal coupling connects integer cluster weights with the Potts model, while random-cluster boundary conditions specify how boundary vertices are identified before component counting.