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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Local limit theorem for symmetric random walks in Gromov-hyperbolic groups
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by Sébastien Gouëzel PDF
J. Amer. Math. Soc. 27 (2014), 893-928 Request permission

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.
References
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Additional Information
  • Sébastien Gouëzel
  • Affiliation: IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042 Rennes, France
  • Email: sebastien.gouezel@univ-rennes1.fr
  • Received by editor(s): September 17, 2012
  • Received by editor(s) in revised form: September 17, 2013
  • Published electronically: March 20, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 27 (2014), 893-928
  • MSC (2010): Primary 05C81, 60J50, 20F67
  • DOI: https://doi.org/10.1090/S0894-0347-2014-00788-8
  • MathSciNet review: 3194496