Two-Dimensional Random Walk

From Path Counting to Random Interlacements

Serguei Popov (Universidade do Porto)

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English

Cambridge University Press
18 March 2021

Calculus & mathematical analysis; Probability & statistics; Stochastics

Series: Institute of Mathematical Statistics Textbooks

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.

By: Serguei Popov (Universidade do Porto)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 228mm, Width: 151mm, Spine: 12mm
Weight: 327g
ISBN: 9781108459693
ISBN 10: 1108459692
Series: Institute of Mathematical Statistics Textbooks
Pages: 200
Publication Date: 18 March 2021
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

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.

Reviews for Two-Dimensional Random Walk: From Path Counting to Random Interlacements

'An excellent and inspiring introduction to simple random walk and random interlacements, in transient and recurrent cases. With its careful and original selection of topics, the reader will soon grasp the general picture and main ideas though to quite advanced material. Each chapter has a great selection of exercises with hints and solutions. This book is primarily designed for self-study, but it can also be used for a graduate course on Markov chains or Poisson processes.' Francis Comets, Universite de Paris