| Abstrakt: |
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 of the diameter, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres (Geom Funct Anal 26(4):1190–1216, 2016. 10.1007/s00039-016-0382-7) from the setting of graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from Sarnak and Xue (Duke Math J 64(1):207–227, 1991), we are able to show that the results apply to congruence subgroups of S L 2 Z and other arithmetic lattices, without relying on the well-known conjecture of Selberg (Proc Symp Pure Math 8:1–15, 1965), thus relaxing the condition on the Laplace spectrum of a surface. 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 (Proc Natl Acad Sci 93(4):1659–1664, 1996), who asked under what general phenomena cutoff exists. [ABSTRACT FROM AUTHOR] |
Full text from SpringerLink