Cutoff on Hyperbolic Surfaces

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 $S{L}_2\left(\mathbb{Z}\right)$ 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.

References in corpus (5)

• Super-Golden-Gates for PU(2)
• Density Theorems for Exceptional Eigenvalues for Congruence Subgroups
• Random walks on Ramanujan complexes and digraphs
• $L^p$ Expander Graphs
• Lower bounds for the measurable chromatic number of the hyperbolic plane

Cited by in corpus (5)

• A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\epsilon$
• Optimal Lifting for the Projective Action of $S{L}_3(Z)$
• Optimal Diophantine Exponents for $\mathrm{SL}(n)$
• Small diameters and generators for arithmetic lattices in ${\mathrm{SL}}_2(\mathbb{R})$ and certain Ramanujan graphs
• Improved computation of fundamental domains for arithmetic Fuchsian groups