大属随机双曲曲面的低音

大亏格闭双曲曲面的第二特征值研究

On second eigenvalues of closed hyperbolic surfaces for large genus

本文研究大亏格闭双曲曲面的第二特征值，证明其与分离曲面为三部分的简单闭多重测地线最短长度之比介于 $1/g^2$ 与常数之间且界最优，并进一步证明随机曲面上该比值与 $1/\ln(g)$ 一致可比。

高亏格典型双曲曲面上扭曲拉普拉斯算子的谱分布

Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus

本文研究高亏格典型双曲曲面上扭曲拉普拉斯算子的谱分布，通过调和形式的小上确界范数估计，证明了谱实部分布的一致Weyl定律。

随机双曲曲面上拉普拉斯算子的行列式

Determinants of Laplacians on random hyperbolic surfaces

该研究针对亏格趋于无穷的紧双曲曲面，分析Laplace-Beltrami算子正则化行列式的特性，证实主流随机曲面模型下该行列式高概率以通用指数呈指数增长，并推导其对数矩的极限结果。

Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus

We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. In this note we show that in this same limit, the energy variance for a typical surface …

大亏格双曲面上 Eisenstein 级数的量子遍历性

Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus

本文给出了有限面积双曲面上量子平均绝对偏差的定量估计，证明了在 Benjamini-Schramm 收敛于双曲平面时 Laplacian 特征函数的去局部化，并指出该结果在 Mirzakhani 随机曲面模型中具有通用性。

大亏格曲面上的混沌现象

Chaos in large genus surfaces

本文概述了Anantharaman与Monk的研究，指出大亏格闭双曲曲面在概率意义上具有最优谱间隙，该成果依托多位学者的基础性工作，深化了对双曲曲面测地流混沌特性的认识。

双曲面的线性规划界限

Linear programming bounds for hyperbolic surfaces

本研究通过线性规划方法为双曲面提供了新的上界和下界，涉及系绳长、kissing 数和谱特性，适用于低亏格和高亏格情况。

Systole and $λ_{2g-2}$ of a hyperbolic surface

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${λ_{2g-2}}(S)$: ${λ_{2g-2}}(S) > 1/4 + {ε_0}(S)$, where ${ε_0}(S) > 0$ is an explicit constant which depends only on the systole of $S$

双曲曲面上旋量场的谱间隙

A Spectral Gap for Spinors on Hyperbolic Surfaces

本文构造了亏格趋于无穷的显式旋量双曲曲面序列，其狄拉克算子具有一致的谱间隙，且这些曲面为算术双曲曲面的覆盖塔。

Short geodesic loops and $L^p$ norms of eigenfunctions on large genus random surfaces

We give upper bounds for $L^p$ norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $g \to +\infty$, we show that random hyperbolic surfaces $X$ with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than $c \log g$ for small enough $c > 0$. This allows us to deduce that the $L^p$ norms of $L^2$ normalised eigenfunctions on $X$ are a $O(1/\sqrt{\log g})$ with high probability in the large genus l…