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Random walks and heat kernels on graphs / Martin T. Barlow.

By: Material type: TextTextSeries: London Mathematical Society lecture note series ; 438.Publication details: Cambridge : Cambridge University Press, [2017]Description: xi, 226 pages : illustrations ; 23 cmISBN:
  • 9781107674424
Subject(s): DDC classification:
  • 511.5 23 B258
Contents:
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.
Summary: 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.
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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 511.5 B258 (Browse shelf(Opens below)) Available 138116
Total holds: 0

Includes bibliographical references and index.

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.

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.

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