Over the years the concept of a random walk has found extensive application in many fields ranging from astronomy and solid state physics to polymer chemistry and biology. The motives for the choice of a random walk model to characterize a particular physical problem are often very different, but the details of the subsequent analysis are closely related. Unfortunately, most of the general treatments of the theory of random walks tend to be of a mathematical sophistication which demands an advanced knowledge of probability theory and measure theory. On the other hand, the generating function formalism developed over the last decade, by EW Montroll, and subsequently by GH Weiss, allows a discussion of the random walk problem which can be understood by most scientists with an average mathematical background. This book attempts to unite the various results and applications on the basis of the generating function technique.

The discussion of examples is not complete and, as it must in any book of this length, is necessarily subjective. Applications to solid state physics and the Ising problem might have been more heavily treated. We would have liked to have spent more time on Brownian motion, diffusion theory, master equations generally, the Fokker-Planck equation, and chemical kinetics, although these areas properly belong to non-equilibrium statistical mechanics or stochastic processes. Other applications which we have merely touched upon are to areas like nuclear magnetic resonance, polyelectrolytes, and the kinetics of the helix coil transition. Again these areas begin to spill over into solid state theory, physical chemistry and molecular biology. However, we hope that we have at least mentioned most of the significant applications, and have cited a sufficient number of key papers so that anyone interested in a particular topic can follow it up and develop the area himself. The mathematical methods which are used for random walk problems quite naturally turn up repeatedly throughout theoretical physics. Indeed some of the more sophisticated problems in random walks-most unsolved-lead on to problems in analysis which are quite extraordinarily deep. While our bias has been towards applications, we hope that the reader more vii