双曲面上的局部魏尔定律与长度最小化回路
Local Weyl law and length-minimising loops on hyperbolic surfaces
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中文总结 AI 辅助
研究大的韦伊 - 彼得森双曲曲面上局部魏尔定律的方差，通过特定方法明确积分相关测试函数，利用米尔扎哈尼工作确定渐近行为，引入长度最小化回路概念并证明其拓扑特征，还介绍了探索回路族。
中文摘要 AI 辅助
我们研究了在固定光滑能量窗口上，对大的韦伊 - 彼得森双曲曲面进行平均时局部魏尔定律的方差。我们的结果与贝里随机波模型的预测一致。我们的方法允许明确积分某些依赖于基于测地回路长度的测试函数，并将它们与自由同伦类中闭测地线的相关长度联系起来。我们利用米尔扎哈尼的工作，通过精确的驻相论证，确定正确的主项和误差项，明确局部魏尔定律方差的渐近行为。此外，我们引入了基于一点的长度最小化测地回路和序列的几何概念。我们证明了它们拓扑结构的完整特征，即它们是简单的。这是我们研究的关键要素，并产生了一个新的简化论证来界定依赖于不同短原始测地回路对长度的余项贡献。为了说明我们结果的一般性，我们进一步引入了一族基于一点的“探索”回路，这可能具有独立的研究价值。
英文摘要
We study the variance of a local Weyl law over a fixed smooth energy window, when averaged over large Weil--Petersson hyperbolic surfaces. Our results are consistent with the predictions of Berry's random wave model. Our approach allows to explicitly integrate certain test functions which depend on lengths of based geodesic loops, and relate them to the associated lengths of the closed geodesics in their free-homotopy class. We thus utilise the work of Mirzakhani, with exact stationary phase arguments, to identify correct main and error terms, making explicit the asymptotic behaviour of the variance of the local Weyl law. Furthermore, we introduce the geometric notion of length-minimising geodesic loops and sequences, based at a point. We prove a complete characterisation of the topology of these, namely that they are simple. This forms a key ingredient in our study, and yields a new streamlined argument to bound the contributions of remainder terms which depend on lengths of pairs of different short primitive geodesic loops. To illustrate the generality of our results, we further introduce a family of "exploring" loops based at a point, which might be of independent interest.
