Two-Dimensional Random Walk

The main subject of this introductory book is simple random walk on the integer
lattice, with special attention to the two-dimensional case. This fascinating
mathematical object is the point of departure for an intuitive and richly illustrated tour
of related topics at the active edge of research. The book starts with three different
proofs of the recurrence of the two-dimensional walk, via direct combinatorial
arguments, electrical networks, and Lyapunov functions. Then, after reviewing some
relevant potential-theoretic tools, the reader is guided towards the relatively new topic
of random interlacements – which can be viewed as a “canonical soup” of
nearest-neighbour loops through infinity – again with emphasis on two dimensions.
On the way, readers will visit conditioned simple random walks – which are the
“noodles” in the soup – and also discover how Poisson processes of infinite objects are
constructed and review the recently introduced method of soft local times. Each
chapter ends with many exercises, making the book suitable for courses and for
independent study.