Two-Dimensional Random Walk 1st Edition Serguei Popov 1108591124 9781108591126

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Product details:

• ISBN 10: 1108591124
• ISBN 13: 9781108591126
• Author: Serguei Popov

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.

Table of contents:

1 Introduction

2 Recurrence of two-dimensional simple random walk

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

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