In 1969, Fortuin and Kasteleyn (FK) [27,28,103,104] found an interesting mapping between the q-state Potts model, which includes the Ising model for q = 2, and a correlated bond-percolation model called the random-cluster model. It can be shown that there is a one-to-one correspondence between different thermodynamic quantities and their geometric counterparts based on the statistical and fractal properties of FK clusters.
This allowed powerful renormalization group ideas to be used [74].
Swendsen and Wang [105], and then Wolff [106], have exploited this mapping to devise extraordinar- ily efficient Monte Carlo algorithms.
There are mappings between the Ising model at a given dimension and a model of manifolds surrounding the geometric spin clusters.
Percolation and the Potts model. Many of the tools of Statistical physics have been applied to percolation through these mappings.