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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Difference equations, isoperimetric inequality and transience of certain random walks


Author: Jozef Dodziuk
Journal: Trans. Amer. Math. Soc. 284 (1984), 787-794
MSC: Primary 58G32; Secondary 35J05, 39A12, 53C99
MathSciNet review: 743744
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Abstract: The difference Laplacian on a square lattice in $ {{\mathbf{R}}^n}$ has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0743744-X
PII: S 0002-9947(1984)0743744-X
Article copyright: © Copyright 1984 American Mathematical Society