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
Cartwright, D., Soardi, P., Woess, W. (1993). Martin and end compactifications for non-locally finite graphs. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 338(2), 679-693 [10.1090/S0002-9947-1993-1102885-3].
Martin and end compactifications for non-locally finite graphs
SOARDI, PAOLO MAURIZIO;
1993
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 generatorsI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.