Introductory remarks

There are many beautiful problems of physical type that may be modelled as Markov processes on the compact state space = {0, 1}V for some countable set V. Amongst the most studied to date by probabilists are the contact, voter, and exclusion models, and the stochastic Ising model.

Subcritical phase

In language borrowed from the theory of branching processes, a percolation process is termed subcritical if p < pc, and supercritical if p > pc.

In the subcritical phase, all open clusters are (almost surely) finite. The chance of a long-range connection is small, and it approaches zero as the distance between the endpoints diverges. The process is considered to be ‘disordered’, and the probabilities of long-range connectivities tend to zero exponentially in the distance. Exponential decay may be proved by elementary means for sufficiently small p, as in the proof of Theorem 3.2, for example. It is quite another matter to prove exponential decay for all p < pc, and this was achieved for percolation by Aizenman and Barsky and Menshikov around 1986.

Random walk – the stochastic process formed by successive summation of independent, identically distributed random variables – is one of the most basic and well-studied topics in probability theory. For random walks on the integer lattice ℤd, the main reference is the classic book by Spitzer (1976). This text considers only a subset of such walks, namely those corresponding to increment distributions with zero mean and finite variance. In this case, one can summarize the main result very quickly: the central limit theorem implies that under appropriate rescaling the limiting distribution is normal, and the functional central limit theorem implies that the distribution of the corresponding path-valued process (after standard rescaling of time and space) approaches that of Brownian motion.

Researchers who work with perturbations of random walks, or with particle systems and other models that use random walks as a basic ingredient, often need more precise information on random walk behavior than that provided by the central limit theorems. In particular, it is important to understand the size of the error resulting from the approximation of random walk by Brownian motion. For this reason, there is need for more detailed analysis. This book is an introduction to the random walk theory with an emphasis on the error estimates. Although “mean zero, finite variance” assumption is both necessary and sufficient for normal convergence, one typically needs to make stronger assumptions on the increments of the walk in order to obtain good bounds on the error terms.

Within the menagerie of objects studied in contemporary probability theory, there are a number of related animals that have attracted great interest amongst probabilists and physicists in recent years. The inspiration for many of these objects comes from physics, but the mathematical subject has taken on a life of its own, and many beautiful constructions have emerged. The overall target of these notes is to identify some of these topics, and to develop their basic theory at a level suitable for mathematics graduates.

If the two principal characters in these notes are random walk and percolation, they are only part of the rich theory of uniform spanning trees, self-avoiding walks, random networks, models for ferromagnetism and the spread of disease, and motion in random environments. This is an area that has attracted many fine scientists, by virtue, perhaps, of its special mixture of modelling and problem-solving. There remain many open problems. It is the experience of the author that these may be explained successfully to a graduate audience open to inspiration and provocation.

The material described here may be used for personal study, and as the bases of lecture courses of between 24 and 48 hours duration. Little is assumed about the mathematical background of the audience beyond some basic probability theory, but students should be willing to get their hands dirty if they are to profit.