Random Walks and Heat Kernels on Graphs

Martin T. Barlow (University of British Columbia, Vancouver)

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English

Cambridge University Press
23 February 2017

Calculus & mathematical analysis; Probability & statistics; Combinatorics & graph theory; Stochastics; Thermodynamics & heat

Series: London Mathematical Society Lecture Note Series

This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincaré inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere.

By: Martin T. Barlow (University of British Columbia Vancouver)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 438
Dimensions: Height: 226mm, Width: 152mm, Spine: 15mm
Weight: 350g
ISBN: 9781107674424
ISBN 10: 1107674425
Series: London Mathematical Society Lecture Note Series
Pages: 236
Publication Date: 23 February 2017
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Preface; 1. Introduction; 2. Random walks and electrical resistance; 3. Isoperimetric inequalities and applications; 4. Discrete time heat kernel; 5. Continuous time random walks; 6. Heat kernel bounds; 7. Potential theory and Harnack inequalities; Appendix A; References; Index.

Martin T. Barlow is Professor in the Mathematics Department at the University of British Columbia. He was one of the founders of the mathematical theory of diffusions on fractals, and more recently has worked on random walks on random graphs. He gave a talk at the International Congress of Mathematicians (ICM) in 1990, and was elected a Fellow of the Royal Society of Canada in 1998 and a Fellow of the Royal Society in 2005. He is the winner of the Jeffrey-Williams Prize of the Canadian Mathematical Society and the CRM-Fields-PIMS Prize of the three Canadian mathematics institutes (the Centre de recherches mathématiques, the Fields Institute, and the Pacific Institute for the Mathematical Sciences).

Reviews for Random Walks and Heat Kernels on Graphs

'This book, written with great care, is a comprehensive course on random walks on graphs, with a focus on the relation between rough geometric properties of the underlying graph and the asymptotic behavior of the random walk on it. It is accessible to graduate students but may also serve as a good reference for researchers. It contains the usual material about random walks on graphs and its connections to discrete potential theory and electrical resistance (Chapters 1, 2 and 3). The heart of the book is then devoted to the study of the heat kernel (Chapters 4, 5 and 6). The author develops sufficient conditions under which sub-Gaussian or Gaussian bounds for the heat kernel hold (both on-diagonal and off diagonal; both upper and lower bounds).' Nicolas Curien, Mathematical Review 'This book, written with great care, is a comprehensive course on random walks on graphs, with a focus on the relation between rough geometric properties of the underlying graph and the asymptotic behavior of the random walk on it. It is accessible to graduate students but may also serve as a good reference for researchers. It contains the usual material about random walks on graphs and its connections to discrete potential theory and electrical resistance (Chapters 1, 2 and 3). The heart of the book is then devoted to the study of the heat kernel (Chapters 4, 5 and 6). The author develops sufficient conditions under which sub-Gaussian or Gaussian bounds for the heat kernel hold (both on-diagonal and off diagonal; both upper and lower bounds).' Nicolas Curien, Mathematical Review