PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 149
The rate of publication of papers on percolation and its ramifications is
very high in the physics journals, although substantial mathematical contributions
are rare. The depth of the ‘culture chasm’ is such that few (if anyone)
can honestly boast to understand all the major mathematical and physical
ideas which have contributed to the subject.
1.3 History
In 1957, Simon Broadbent and John Hammersley [80] presented a model for
a disordered porous medium which they called the percolation model. Their
motivation was perhaps to understand flow through a discrete disordered
system, such as particles flowing through the filter of a gas mask, or fluid
seeping through the interstices of a porous stone. They proved in [80, 175,
176] that the percolation model has a phase transition, and they developed
some technology for studying the two phases of the process.
These early papers were followed swiftly by a small number of high quality
articles by others, particularly [137, 182, 338], but interest flagged for
a period beginning around 1964. Despite certain appearances to the contrary,
some individuals realised that a certain famous conjecture remained
unproven, namely that the critical probability of bond percolation on the
square lattice equals 1
2 . Fundamental rigorous progress towards this conjecture
was made around 1976 by Russo [325] and Seymour and Welsh [337], and
the conjecture was finally resolved in a famous paper of Kesten [201]. This
was achieved by a development of a sophisticated mechanism for studying
percolation in two dimensions, relying in part on path-intersection properties
which are not valid in higher dimensions. This mechanism was laid out more
fully by Kesten in his monograph [203].
Percolation became a subject of vigorous research by mathematicians and
physicists, each group working in its own vernacular. The decade beginning
in 1980 saw the rigorous resolution of many substantial difficulties, and the
formulation of concrete hypotheses concerning the nature of phase transition.
The principal progress was on three fronts. Initially mathematicians concentrated
on the ‘subcritical phase’, when the density p of open edges satisfies
p < pc (here and later, pc denotes the critical probability). It was in this context
that the correct generalisation of Kesten’s theorem was discovered, valid
for all dimensions (i.e., two or more). This was achieved independently by
Aizenman and Barsky [12] and Menshikov [267, 268].
The second front concerned the ‘supercritical phase’, when p > pc. The
key question here was resolved by Grimmett and Marstrand [164] following
work of Barsky, Grimmett, and Newman [49].
The critical case, when p is near or equal to the critical probability pc,
remains largely unresolved by mathematicians (except when d is sufficiently
large). Progress has certainly been made, but we seem far from understanding
the beautiful picture of the phase transition, involving scaling theory and
renormalisation, which is displayed before us by physicists. This multifaceted