Cutoff on hyperbolic surfaces

We consider a constant length step random walk on a hyperbolic surface, and deduce that the walker eventually gets lost (i.e., converges to the uniform distribution), and under the assumption of optimality of the non-trivial Laplace spectrum on the surface, the walker gets lost suddenly (i.e., the walk exhibits cut-off). We also prove that under the assumption of optimality the distances between pair of points of the surface are highly concentrated.

Analogous results were proved for graphs by Lubetzky and Peres, and for simplicial complexes by Lubetzky, Lubotzky and Parzanchevski. We show that conceptually the results in all three settings are closely related to the temperedness of representations of corresponding algebraic groups.

Joint work with Amitay Kamber [https://arxiv.org/abs/1712.10149]