Journal of the European Mathematical Society
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Volume 5, Issue 4, 2003, pp. 313–342
DOI: 10.1007/s10097-003-0054-4
Random walks on finite rank solvable groups
Christophe Pittet (1) and Laurent Saloff-Coste (2)
(1) Laboratoire de Mathématiques Emile Picard, Université Paul Sabatier, 118 route de Narbonne, F-31062, TOULOUSE CEDEX, FRANCE(2) Department of Mathematics, Cornell University, 310 Malott Hall, NY 14853-4201, ITHACA, UNITED STATES
We establish the lower bound p2t(e,e)scapexp(-t1/3), for the large times asymptotic behaviours of the probabilities p2t(e,e) of return to the origin at even times 2t, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r, such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to r.)
Keywords: random walk, heat kernel decay, asymptotic invariants of infinite groups, Prüfer rank, solvable group