Riemann surfaces随机覆盖的显式谱间隙
Explicit spectral gaps for random covers of Riemann surfaces
摘要：
本文引入了非初等凸余紧双曲曲面$X=Γ\backslash\mathbb{H}$的随机度$n$覆盖$X_{n}$的置换模型。令$δ$是$Γ$的极限集的Hausdorff维数。如果$X_{n}$的共振不是$X$的共振，我们说它是新的，并且类似地定义拉普拉斯算子的新特征值。我们证明，对于任何$ε>0$和PDMATHAMT，概率倾向于$1$作为$n\to\infty$，有没有新的共振$s=σ+it$的$X_{n}$与$σ\in[\frac{3}{4}δ+ε,δ]$和$t\in[-H,H]$。这意味着在$δ>\frac{1}{2}$的情况下，存在一个显式区间，其中$X_{n}$上没有拉普拉斯算子的新特征值。通过结合这些结果与一个确定性的“高频”无共振带的结果，我们得到的推论是，有一个$η=η(X)$，使概率$\to1$为$n\to\infty$，有没有新的共振$X_{n}$在区域$\{\,s\,:\,\mathrm{Re}(s)>δ-η\,\}$。
英文摘要：
We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=Γ\backslash\mathbb{H}$. Let $δ$ be the Hausdorff dimension of the limit set of $Γ$. We say that a resonance of $X_{n}$ is new if it is not a resonance of $X$, and similarly define new eigenvalues of the Laplacian. We prove that for any $ε>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=σ+it$ of $X_{n}$ with $σ\in[\frac{3}{4}δ+ε,δ]$ and $t\in[-H,H]$. This implies in the case of $δ>\frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$. By combining these results with a deterministic `high frequency' resonance-free strip result, we obtain the corollary that there is an $η=η(X)$ such that with probability $\to1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>δ-η\,\}$.
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