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Random Walks and Heat Kernels on Graphs
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  • Cited by 3
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    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Dembo, Amir Huang, Ruojun and Zheng, Tianyi 2018. Random walks among time increasing conductances: heat kernel estimates. Probability Theory and Related Fields,

    Weihrauch, Tobias 2018. A Characterization of Effective Resistance Metrics. Potential Analysis,

    Haeseler, Sebastian Huang, Xueping Lenz, Daniel and Pogorzelski, Felix 2017. Diffusion on Delone sets. Journal of Statistical Physics, Vol. 167, Issue. 6, p. 1496.

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Book description

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.

Reviews

'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 Source: Mathematical Review

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