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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Martin and end compactifications for non-locally finite graphs

Authors: Donald I. Cartwright, Paolo M. Soardi and Wolfgang Woess
Journal: Trans. Amer. Math. Soc. 338 (1993), 679-693
MSC: Primary 60J15; Secondary 60J50
MathSciNet review: 1102885
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Abstract: We consider a connected graph, having countably infinite vertex set $ X$, which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix $ P$ corresponding to a nearest neighbor random walk on $ X$, we study the associated harmonic functions on $ X$ and, in particular, the Martin compactification. We also study the end compactification of the graph. When the graph is a tree, we show that these compactifications coincide; they are a disjoint union of $ X$, the set of ends, and the set of improper vertices--new points associated with vertices of infinite degree. Other results proved include a solution of the Dirichlet problem in the context of the end compactification of a general graph. Applications are given to, e.g., the Cayley graph of a free group on infinitely many generators.

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Keywords: Martin boundary, ends, harmonic functions on graphs, Fatou theorem, Dirichlet problem
Article copyright: © Copyright 1993 American Mathematical Society