AI 中文总结
研究紧致流形上拉普拉斯本征函数节点体积丘成桐猜想的概率对应问题，通过研究黎曼随机波高频方差渐近性，建立节点体积波动定量界，改进现有结果，证明贝里抵消现象，依赖多种分析方法并引入通用机制确保方差衰减。
AI 中文摘要
我们通过研究黎曼随机波的高频方差渐近性，来探究紧致流形上拉普拉斯本征函数节点体积的丘成桐猜想的概率对应问题。我们建立了（定理A）节点体积波动的定量界，它取决于谱窗的不同状态，包括单色谱窗（谱大小为1）。在没有共轭对的流形，特别是负曲率流形的情况下，我们的界比文献（Canzani和Hanin，2020）中的现有结果改进了不止2次幂。作为推论，我们证明了在这种混沌流形上，单色黎曼随机波会出现贝里抵消现象。我们的证明依赖于局部和全局分析，结合了卡茨 - 赖斯公式、Stecconi和Todino（2025）的新维纳 - 伊藤混沌分解，以及对与任意谱窗相关的逐点魏尔定律误差的精确分析（定理B）。我们引入了一种通用机制（定理C），在相关衰减假设下，确保在广泛的几何条件下方差衰减。
英文摘要
We investigate the probabilistic counterpart of Yau's conjecture on the nodal volume of Laplace eigenfunctions on compact manifolds, by studying the high-frequency variance asymptotics of Riemannian random waves. We establish (Theorem A) a quantitative bound for the fluctuations of their nodal volumes, depending on different regimes of spectral windows, including the monochromatic one, of spectral size 1. Notably, our bounds improve, by more than a power 2, the existing results in the literature, cf. Canzani and Hanin (2020), in the case of manifolds without conjugate points, in particular negatively curved ones. As a corollary, we prove that Berry's cancellation phenomenon occurs for monochromatic Riemannian Random Waves on such chaotic manifolds. Our proofs rely on a local and global analysis combining the Kac-Rice formula, the new Wiener-Itô chaos decomposition of Stecconi and Todino (2025), and a sharp analysis of the error in the pointwise Weyl law associated to an arbitrary spectral window (Theorem B). We introduce a general machinery (Theorem C), which ensures variance decay under broad geometric conditions, subject to correlation decay assumptions.
