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Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

Author: Sébastien Gouëzel
Journal: J. Amer. Math. Soc. 27 (2014), 893-928
MSC (2010): Primary 05C81, 60J50, 20F67
Published electronically: March 20, 2014
MathSciNet review: 3194496
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Abstract: Completing a strategy of Gouëzel and Lalley, we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non-elementary Gromov-hyperbolic group: denoting by $ R$ the inverse of the spectral radius of the random walk, the probability to return to the identity at time $ n$ behaves like $ C R^{-n}n^{-3/2}$. An important step in the proof is to extend Ancona's results on the Martin boundary up to the spectral radius: we show that the Martin boundary for $ R$-harmonic functions coincides with the geometric boundary of the group. In Appendix A, we explain how the symmetry assumption of the measure can be dispensed with for surface groups.

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Additional Information

Sébastien Gouëzel
Affiliation: IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042 Rennes, France

Keywords: Local limit theorem, random walk, hyperbolic group, spectral radius, Martin boundary, transfer operator
Received by editor(s): September 17, 2012
Received by editor(s) in revised form: September 17, 2013
Published electronically: March 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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