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 $\widetilde C_2$.

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拉马努金复形和有向图上的随机游走

第一作者和佩雷斯最近证实了拉马努金图上的随机游走的截止现象。在这项工作中，我们获得了更高维度的类似物，用于与局部场 $F$ 上的简单组 $G$ 相关联的任何拉马努金复数上的随机游走算子。我们证明，如果 $T$ 是具有简单组合属性（无碰撞）的 Bruhat-Tits 建筑物上的任何 $k$-regular $G$-equivariant 算子，则 $n$-顶点 Ramanujan 复合体上的相关随机游走在时间 $\log_k n$ 处截止。与图不同，高维情况需要来自非交换调和分析和 $G$ 的无限维表示理论的工具。通过这些，我们证明了上述拉马努金复合体上的算子 $T$ 产生了具有特殊性质 ($r$-normal) 的拉马努金有向图，这意味着截止。