Random Walks and Heat Kernels on Graphs

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).

• 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.

'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