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We obtain orbital counting results for the class of strongly hyperbolicmetrics on hyperbolic groups. To achieve this we combine ergodic theoretictechniques involving the Mineyev topological flow and symbolic dynamics. Ourresults apply to the Green metric associated to an admissible, finitelysupported, symmetric random walk and to the Mineyev hat metric. We alsodescribe the domain of analyticity for the Poincar\'e series associated tothese metrics, prove mixing results for the Mineyev topological flow and obtaincorrelation asymptotics for pairs of metrics.

Mixing of the Mineyev flow, orbital counting and Poincar\'e series for strongly hyperbolic metrics