Random walks on Ramanujan complexes and digraphs

Abstract

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group G over a local field F . We show that if T is any k-regular G-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the n-vertex Ramanujan complex has cutoff at time log k n. The high dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of G. Via these, we show that operators T as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property (r-normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group G, previously known for groups of type An and C2.

Cutoff on Ramanujan complexes and classical groups

“…In higher dimension, NBRW is not collision-free anymore, but in [LLP17, §5.1] it is shown that collision-free walks do exist, on cells of every dimension, except for vertices. As SRW is not collision-free, the techniques of [LLP17] cannot address it (in fact, they cannot address any operator on vertices -see [Par19, Rem. 3.5(b)]).…”

“…Most notably, it was shown by Lubetzky and Peres that Ramanujan graphs exhibit SRW cutoff [LP16], and a main ingredient of the proof is to show first that non-backtracking random walk (NBRW) on these graphs exhibits cutoff at an optimal time. The last assertion was generalized in [LLP17] to the context of Ramanujan complexes, which are high-dimensional analogues of Ramanujan graphs defined in [Li04,LSV05a]. In the paper [LLP17], Lubetzky, Lubotzky and the second author establish optimal-time cutoff for a large family of asymmetric random walks on the cells of these complexes (in the graph case, NBRW is an asymmetric walk on edges).…”

“…The last assertion was generalized in [LLP17] to the context of Ramanujan complexes, which are high-dimensional analogues of Ramanujan graphs defined in [Li04,LSV05a]. In the paper [LLP17], Lubetzky, Lubotzky and the second author establish optimal-time cutoff for a large family of asymmetric random walks on the cells of these complexes (in the graph case, NBRW is an asymmetric walk on edges). However, the techniques of [LLP17] can not be applied to any symmetric random walk, and in particular to SRW on vertices.…”

“…In the paper [LLP17], Lubetzky, Lubotzky and the second author establish optimal-time cutoff for a large family of asymmetric random walks on the cells of these complexes (in the graph case, NBRW is an asymmetric walk on edges). However, the techniques of [LLP17] can not be applied to any symmetric random walk, and in particular to SRW on vertices. The goal of the current paper is to establish cutoff for SRW on Ramanujan complexes of type A d , namely, the ones which arise from the group P GL d over a non-archimedean local field.…”

“…In accordance, Ramanujan complexes are roughly defined as finite complexes which spectrally mimic their universal cover; for a precise definition see §2. 1 In [LLP17], a vast generalization of part (2) of the theorem above is proved: say that a walk rule is collision-free if two walkers which depart from each other can never cross paths again.…”