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Random walks and random environments. Vol. 2: Random environments. (English) Zbl 0925.60076
Oxford Science Publications. Oxford: Clarendon Press. xxiv, 526 p. (1996).
This volume, apart from basic results of probability theory, self-avoiding walk theory and fractal notations, is reasonably self-contained, and covers both the rigorous and the heuristic sides of percolation theory as well as the theory of transport processes in random media. Chapter 1 of this volume gives an overview of the standard percolation models (known as Bernoulli percolation models). In Chapter 2 some general results for the percolation are proved rigorously, which hold without detailed restrictions on the lattice structure. One of the most important proofs is that a number of critical probabilities coincide and thus a well-defined percolation threshold can be introduced. Chapter 3 discusses the values of the percolation threshold, covering exact and rigorous results in very few cases, and numerical estimates or speculations for Bernoulli percolation models or other models. The statistical behavior of a random system near the percolation threshold is studied in Chapter 4, which is naturally related or analogous to the critical phenomena in statistical mechanics, particularly via the critical exponents and their scaling relations. Chapter 5 concerns the steady-state transport process in random environments, focussing on the random resistor network problem and the percolation conduction problem. Chapter 6 discusses a time-evolving random process taking place in a random environment (mainly restricted to one spatial dimension), while Chapter 7, the last chapter of this volume, is devoted to problems of a random walk on a random lattice created by a percolation process referred to as ”the ant in the labyrinth”. All the material in the book is care fully chosen and reorganized from the huge literature on the subject. The origin of the problems and the physical background and applications of the models are always described in detail. The book also provides an up-to-date view of the most challenging problems in the subject. Due to the elegant and beautiful presentation, this book can be heartily recommended to graduate students and physicists in related fields, and even experienced mathematicians will certainly learn something new from this book.

60G50 Sums of independent random variables; random walks
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43 Percolation
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics