The random cluster model

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The random-cluster model was introduced by Cees Fortuin and Piet Kasteleyn around 1969 as a unification of percolation, Ising, and Potts models, aiming to reconcile the series and parallel laws governing such systems. This innovation sparked a study in stochastic geometry, revealing intricate structures and becoming essential in addressing one of the longstanding challenges in classical statistical mechanics: modeling and analyzing ferromagnetism and its phase transitions. However, the model's significance in probability and statistical mechanics wasn't fully acknowledged until the late 1980s, primarily due to two factors. First, early publications from 1969 to 1972 emphasized combinatorial aspects, which may have obscured its broader applications. Second, many necessary geometrical arguments for the model's study were not developed until the 1980s during the so-called ‘decade of percolation.’ A pivotal moment came in 1980 with the proof concerning bond percolation on the square lattice, followed by Harry Kesten’s monograph on t-dimensional percolation. By 1986, percolation theories expanded into higher dimensions, resolving many contemporary mathematical challenges. Renewed interest in the random-cluster model as a tool for exploring Ising/Potts models emerged around 1987.