Since the term "random field'' has a variety of different connotations, ranging from
agriculture to statistical mechanics, let us start by clarifying that, in this book, a
random field is a stochastic process, usually taking values in a Euclidean space, and
defined over a parameter space of dimensionality at least 1.
Consequently, random processes defined on countable parameter spaces will not
appear here. Indeed, even processes on R1 will make only rare appearances and,
from the point of view of this book, are almost trivial. The parameter spaces we like
best are manifolds, although for much of the time we shall require no more than that
they be pseudometric spaces.
With this clarification in hand, the next thing that you should know is that this
book will have a sequel dealing primarily with applications.
In fact, as we complete this book, we have already started, together with KW
(Keith Worsley), on a companion volume  tentatively entitled RFG-A, or Random
Fields and Geometry: Applications. The current volume--RFG--concentrates on
the theory and mathematical background of random fields, while RFG-A is intended
to do precisely what its title promises. Once the companion volume is published,
you will find there not only applications of the theory of this book, but of (smooth)
random fields in general.
Making a clear split between theory and practice has both advantages and disad-