TABLE OF CONTENTS We begin this section with a brief indication of how CFTP fits in to the theory of MCMC. We then discuss one of the simplest possible examples of coupling ( 1.1), before describing classic CFTP as applied to the doubly-reflecting random walk ( 1.2). This serves as introduction to the fundamental theorem of CFTP ( 1.3), which is further illustrated by two simple applications: to the dead leaves model ( 1.4) and to the Ising model ( 1.5). The section is completed by a rather less trivial application to point processes ( 1.6) and a discussion of CFTP in space and time ( 1.7), and finally a brief note on some historical and other complementary aspects of CFTP ( 1.8).
MCMC arises in a number of different areas of mathematical science, with different emphases. (This makes interaction interesting and fruitful!) Here are some examples, several of which are discussed at length in other chapters in this volume:
Statistical mechanics. Are there phase transition phenomena in specific infinite-dimensional systems? How do they behave?
Computer science. Approximate counting problems can be solved in terms of algorithms which deliver approximately uniform random samples, which in turn can be solved using MCMC. In this area the key question is, how does the algorithm behave as the scale of the problem increases? Does the run-time increase exponentially, or polynomially?
Image analysis. Given a noisy picture with some kind of geometric content: can we clean it up...
