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Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. (English) Zbl 1320.60017
Summary: Completing a strategy of S. Gouëzel and S. P. Lalley [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 1, 129–173 (2013; Zbl 1277.60012)], 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 [A. Ancona, Ann. Math. (2) 125, 495–536 (1987; Zbl 0652.31008)]: 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.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
05C81 Random walks on graphs
60J50 Boundary theory for Markov processes
20F67 Hyperbolic groups and nonpositively curved groups
31C35 Martin boundary theory
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