Spectral statistics for random hyperbolic surfaces

本研究项目旨在证明20世纪80年代在量子混沌背景下做出的有影响力的猜想，包括混沌量子系统谱统计的Bohigas-Giannoni-Schmit （BGS）猜想和随机矩阵理论。在过去的三十年中，物理学文献中的大量工作通过混沌经典粒子轨迹的相关性提供了随机矩阵统计的启发式解释，我们将重点关注量子混沌的特别干净的数学模型-双曲表面上的拉普拉斯。双曲曲面的研究是有趣的，因为它们提供了丰富的例子与混沌动力学（由于负曲率），并允许应用强大的数学工具。这个项目的新颖之处在于使用遍历理论中最近发展的技术来解决一些突出的平均猜想，也就是说，不是针对单个固定曲面（在那里挑战太难了），而是通过在给定属的曲面的模空间上取平均值。使用Selberg迹公式（该学科的标准工具），光谱统计量将映射到封闭测地线长度的几何相关性，分析中的关键挑战将是证明随机表面上封闭测地线分布的严格极限定理。在这里，我们将利用几何学家最近令人兴奋的突破，包括菲尔兹奖得主米尔扎哈尼和其他人。

This research project aims to make progress towards the proof of influential conjectures made in the 1980s in the context of quantum chaos, including the Bohigas-Giannoni-Schmit (BGS) conjecture on spectral statistics of chaotic quantum systems and random matrix theory. Following a large body of work in the physics literature over the past three decades, which provided a heuristic explanation of random matrix statistics through correlations of chaotic classical particle trajectories, we will focus on particularly clean mathematical models of quantum chaos -- the Laplacian on hyperbolic surfaces. The study of hyperbolic surfaces is interesting because they provide a rich family of examples with chaotic dynamics (due to the negative curvature) and allow the application of powerful mathematical tools. The novelty of this project is to use recently developed techniques in ergodic theory to address some of the outstanding conjectures on average, that is, not for a single fixed surface (where the challenges are simply too hard) but by taking the mean over the moduli space of surfaces of a given genus. Using the Selberg trace formula, a standard tool in the subject, the spectral statistics will be mapped to geometric correlations of lengths of closed geodesics, and the key challenge in the analysis will be to prove rigorous limit theorems for the distribution of closed geodesics on random surfaces. Here we will exploit recent exciting breakthroughs by geometers including Fields medalist Mirzakhani and others.