Grimmett's book, Percolation, is excellent. Percolation theory began in the 50's; its mathematics is now quite mature, but the theory has recently acquired new techniques because many of the questions initially raised by percolation theory are still unanswered.
Percolation technology is now a cornerstone of the theory of disordered systems, and the methods of this book are now being extended into dynamical systems theory and the life sciences. This book covers the mathematics of percolation theory, presenting the shortest rigorous proofs of the main facts. Many problems in percolation theory are beautiful, but some of the apparent simplicity of the subject is deceiving, because the subject is quite deep. Grimmett cuts through many of the difficulties presenting the important concepts clearly and sucinctly.
The author restricts himself- for accessibility to the maximum readership-to bond percolation on a cubic lattice. Grimmett presents the core material at a graduate level for folks conversant with elementary probability theory and real analysis. Having some knowledge of ergodic theory, graph theory, and some mathematical physics helps, however. There is litle discussion of continuous, mixed, inhomogenous, long range, first passage or oriented percolation.
Beginning with existance of Psubc for the edge probability p we arrive at an infinite open cluster followed by discusssion of the basic techniques of the FKC, BK inequalities and Russo's formula. Grimmett then discusses open clusters per vertex and subcritical percolation, beginning with the Aizeman-Barsky and Menshikov methods for identifying the critical point, followed by a systematic study of the subcritical phase. He then discusses supercritical percolation, including 2 dimensional percolation, continuum percolation and random processes. The author gives a full list of references.
