Geodesic flow ergodicity on infinite Riemann surfaces equals recurrence of the random walk on the dual pants graph.
Nicholls,The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, 143
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Quadratic differentials and random walks on the dual graph of a pants decomposition
Geodesic flow ergodicity on infinite Riemann surfaces equals recurrence of the random walk on the dual pants graph.