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arXiv · 1712.10149

Cutoff on Hyperbolic Surfaces

Konstantin Golubev·Amitay Kamber
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Abstract

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal possible value, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres ([20]) from the setting of Ramanujan graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from [27], we are able to show that the results apply to congruence subgroups of $SL_{2}\left(\mathbb{Z}\right)$ and other arithmetic lattices, without relying on the well known conjecture of Selberg ([28]). Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis ([7]), who asked under what general phenomena cutoff exists.

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BibTeXRIS

Konstantin Golubev, Amitay Kamber. 2017-12-29. Cutoff on Hyperbolic Surfaces. https://doi.org/10.1007/s10711-019-00433-5

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