Growth Series and Random Walks on Some Hyperbolic Graphs |
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Authors: | Laurent Bartholdi and Tullio G Ceccherini-Silberstein |
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Institution: | (1) Department of Mathematics, Evans Hall, University of California, CA, 94720-3840 Berkeley, U.S.A.;(2) Facoltà di Ingegneria, Università del Sannio, Palazzo dell'Aquila Bosco Lucarelli, Corso Garibaldi, 82100 Benevento, Italy |
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Abstract: | Consider the tessellation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of proper loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs. |
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