Spectral statistics for random hyperbolic surfaces

随机双曲曲面的谱统计

基本信息

批准号：
EP/W007010/1
负责人：
Jens Marklof
金额：
$ 45.97万
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW007010%2F1
关键词：
Spectral statistics random hyperbolic surfaces

项目摘要

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.

该研究项目旨在在量子混乱的背景下朝着1980年代做出的有影响力的猜想做出进展，包括关于混乱量子系统和随机矩阵理论的频谱统计的Bohigas-Giannoni-Schmit（BGS）猜想。在过去三十年中，在物理文献中进行了大量作品，这通过混乱的古典粒子轨迹的相关性提供了对随机矩阵统计的启发式解释，我们将专注于量子混乱的特别干净的数学模型 - laplacian在双层表面上的laplacian。双曲线表面的研究很有趣，因为它们为混乱的动力学提供了丰富的示例（由于负曲率），并允许应用强大的数学工具。该项目的新颖性是要使用千古理论中最近开发的技术来解决一些杰出的猜想，也就是说，对于单个固定表面而言（挑战太难太难了），而是通过在给定属的表面模量上进行平均值。使用Selberg Trace公式（该主题中的标准工具），光谱统计数据将映射到封闭的大地测量学长度的几何相关性，分析的主要挑战将是证明对随机表面上封闭的大地测量学分布的严格限制定理。在这里，我们将利用包括田野奖得主Mirzakhani等的几何图形的最新突破。