Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs

Abstract

We study the connections between volume growth, spectral properties and stochastic completeness of locally finite graphs. For a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum. We then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite graphs.

29 pages, 2 figures. Some references and definition environments added to the new version. Because of this, Lemmas 3.2 through 3.5 from Section 3 in the previous version are now Lemmas 3.3 through 3.6. Likewise, Lemmas 4.3 and 4.4 became Lemmas 4.4 and 4.5 and Corollary 5.2 is now Corollary 5.3 in the new version. Final version to appear in Math. Z