T h e theory of percolation models is developed following general ideas in the area of critical phenomena. The review is an exposition of current phase transition theory in a geometrical context. As such, it includes a discussion of scaling relations between critical exponents and their calculation using series expansion methods. Renormalisation group techniques are also considered.The major difference between percolation and other phase transition models is the absence of a Hamiltonian. Instead, the theory is based entirely on probabilistic arguments. A discussion of the connections with classical probability theory is also given.

An exact method for determining the critical percolation probability, pc, for a number of two-dimensional site and bond problems is described. For the site problem on the plane triangular lattice pc = ½. For the bond problem on the triangular, simple quadratic, and honeycomb lattices, pc=2 sin (118π),12,1−2 sin (118π), respectively. A matching theorem for the mean number of finite clusters on certain two-dimensional lattices, somewhat analogous to the duality transformation for the partition function of the Ising model, is described.

The problem of cluster size distribution and percolation on a regular lattice or graph of bonds and sites is reviewed and its applications to dilute ferromagnetism, polymer gelation, etc., briefly discussed. The cluster size and percolation problems are then solved exactly for Bethe lattices (infinite homogeneous Cayley trees) and for a wide class of pseudolattices derived by replacing the bonds and/or sites of a Bethe lattice by arbitrary finite subgraphs. Explicit expressions are given for the critical probability (density), for the mean cluster size, and for the density of infinite clusters. The nature of the critical anomalies is shown to be the same for all lattices discussed; in particular, the density of infinite clusters vanishes as R(p) ≈ C(p−pc) (p≥pc).

Abstract. Exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice. The corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained. The work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice. A duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve. The moments of both distributions have critical exponents which satisfy a c o t " gap hypothesis with gap exponents U,-, = 2 and A = 3 corresponding to the scaling length and scaling size respectively. The usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced bywhere U, is the exponent corresponding to the cluster width and D is the number of transverse dimensions ( = 1 for the square lattice).

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