论文标题
高属的随机双曲表面的Benjamini-Schramm收敛和光谱
Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus
论文作者
论文摘要
我们研究了高属的典型双曲表面的几何和光谱特性，不包括针对Weil-Petersson概率度量的一组小量度。我们首先证明Benjamini-Schramm收敛到双曲机h，因为G属G属无穷大。然后使用Selberg痕量公式证明了以A，B和G为间隔[A，B]中特征值数量的估计值。这意味着光谱量度与h的光谱度量的融合为g $ \ rightarrow $+$ \ infty $，以及统一的weyl Law作为b $ \ rightarrow $+$+$ \ infty $。我们推断出小的特征值的数量以及任何特征值的多重性。
We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as the genus g goes to infinity. An estimate for the number of eigenvalues in an interval [a,b] in terms of a, b and g is then proven using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of H as g $\rightarrow$+$\infty$, and a uniform Weyl law as b $\rightarrow$+$\infty$. We deduce a bound on the number of small eigenvalues, and the multiplicity of any eigenvalue.
