Therandom-clustermodelwasinventedbyCees[Kees]FortuinandPietKasteleyn
around 1969 as a uni?cation of percolation, Ising, and Potts models, and
as an extrapolation of electrical networks. Their original motivation
was to harmonize the series and parallel laws satis?ed by such systems.
In so doing, they initiated a study in stochastic geometry which has
exhibited beautiful structure in its own right, and which has become a
central tool in the pursuit of one of the oldest challenges of classical
statistical mechanics, namely to model and analyse the ferromagnet and
especially its phase transition. The importance of the model for
probability and statistical mechanics was not fully recognized until the
late 1980s. There are two reasons for this period of dormancy. Although
the early publications of 1969-1972 contained many of the basic
properties of the model, the emphasis placed there upon combinatorial
aspects may have obscured its potential for applications. In addition,
many of the geometrical arguments necessary for studying the model were
not known prior to 1980, but were developed during the 'decade of
percolation' that began 1 then. In 1980 was published the proof that p =
for bond percolation on the c 2 square lattice, and this was followed
soon by Harry Kesten's monograph on t- dimensionalpercolation.
Percolationmovedintohigherdimensionsaround1986, and many of the
mathematical issues of the day were resolved by 1989. Interest in the
random-cluster model as a tool for studying the Ising/Potts models was
rekindled around 1987.
