Two-Dimensional Random Walk
From Path Counting to Random Interlacements

A visual, intuitive introduction in the form of a tour with side-quests, using direct probabilistic insight rather than technical tools.

Serguei Popov (Author)

9781108459693, Cambridge University Press

Paperback / softback, published 18 March 2021

200 pages
22.8 x 15.1 x 1.2 cm, 0.327 kg

'… a well-written summary of the subject … Highly recommended.' M. Bona, Choice Connect

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. It starts with three different proofs of the recurrence of the two-dimensional walk, via direct combinatorial arguments, electrical networks, and Lyapunov functions. After reviewing some relevant potential-theoretic tools, the reader is guided toward 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 it suitable for courses and independent study.

1. Introduction
2. Recurrence of Two-Dimensional SRW
3. Some Potential Theory for Simple Random Walks
4. SRW Conditioned on not Hitting the Origin
5. Intermezzo: Soft Local Times and Poisson Processes of Objects
6. Random Interlacements.

Subject Areas: Stochastics [PBWL], Probability & statistics [PBT], Calculus & mathematical analysis [PBK]