Percolation on fuchsian groups
Random walks on co-compact fuchsian groups
It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence . It is also shown that Ancona’s inequalities extend to , and therefore that the Martin boundary for -potentials coincides with the natural geometric boundary , and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, .
Strict convexity of the limit shape in first-passage percolation.
An oriented competition model on .
Geometric interpretation of half-plane capacity.
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