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A random walk on a separable, geodesic hyperbolic metric space converges to the boundary with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when (1) the step distribution has exponential tail and (2) the action on is acylindrical.
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Matthew H. Sunderland, Linear progress with exponential decay in weakly hyperbolic groups. Groups Geom. Dyn. 14 (2020), no. 2, pp. 539–566DOI 10.4171/GGD/554