Martin and end compactifications for non-locally finite graphs
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- by Donald I. Cartwright, Paolo M. Soardi and Wolfgang Woess
- Trans. Amer. Math. Soc. 338 (1993), 679-693
- DOI: https://doi.org/10.1090/S0002-9947-1993-1102885-3
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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.References
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Bibliographic Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 338 (1993), 679-693
- MSC: Primary 60J15; Secondary 60J50
- DOI: https://doi.org/10.1090/S0002-9947-1993-1102885-3
- MathSciNet review: 1102885