Geodesic
Local limit theorem for symmetric random walks in Gromov-hyperbolic groups
Gouëzel, Sébastien 1
1 IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042 Rennes, France
Journal of the American Mathematical Society, Tome 27 (2014) no. 3, pp. 893-928 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S0894-0347-2014-00788-8

Gouëzel, Sébastien  1

1 IRMAR, CNRS UMR 6625, Université de Rennes 1, 35042 Rennes, France 
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Gouëzel, Sébastien. Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. Journal of the American Mathematical Society, Tome 27 (2014) no. 3, pp. 893-928. doi: 10.1090/S0894-0347-2014-00788-8

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