局部环上的自由旗与高维扩张子的幂运算
Free flags over local rings and powering of high dimensional expanders
AI总结：
本文针对高维扩张子提出基于 Bruhat–Tits 建筑商空间测地游走的新幂运算，并通过有限局部环自由模旗与 Garland 技术证明其保持并提升扩张性，还构造了高效双重采样器。
摘要：
对扩张图的邻接矩阵取幂，会得到一个次数更高、性能更好的扩张子。本文寻求高维扩张子的类似运算。我们证明朴素的取幂方法不能保持高维扩张性，并利用 Bruhat–Tits 建筑商空间上的测地游走定义了一种新的幂运算。应用该运算可得到次数更高的高维扩张子。证明的关键是对有限局部环上自由模的旗进行组合研究。它们的几何结构描述了幂复形中的链接；证明这些链接是极佳扩张子后，即可通过 Garland 的局部到整体技术推出幂复形的高维扩张性。作为应用，我们利用该幂运算获得了新的高效双重采样器。
英文摘要：
Powering the adjacency matrix of an expander graph results in a better expander of higher degree. In this paper we seek an analogue operation for high-dimensional expanders. We show that the naive approach to powering does not preserve high-dimensional expansion, and define a new power operation, using geodesic walks on quotients of Bruhat-Tits buildings. Applying this operation results in high-dimensional expanders of higher degrees. The crux of the proof is a combinatorial study of flags of free modules over finite local rings. Their geometry describes links in the power complex, and showing that they are excellent expanders implies high dimensional expansion for the power-complex by Garland's local-to-global technique. As an application, we use our power operation to obtain new efficient double samplers.
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