# Random walks on Ramanujan complexes and digraphs

## TL;DR

This paper extends the cutoff phenomenon for random walks from Ramanujan graphs to higher-dimensional Ramanujan complexes, using advanced harmonic analysis and representation theory, and confirms related conjectures including the Riemann Hypothesis for associated zeta functions.

## Contribution

It introduces a higher-dimensional analog of the cutoff phenomenon for random walks on Ramanujan complexes, employing non-commutative harmonic analysis and representation theory.

## Key findings

- Cutoff at time log_k n for certain random walks on Ramanujan complexes.
- Operators on complexes produce Ramanujan digraphs with r-normal property.
- Confirmation of the Riemann Hypothesis for zeta functions over various groups.

## 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 $\widetilde A_n$ and $\widetilde C_2$.