论文标题
紧凑双曲表面的随机盖有相对的光谱差距$ \ frac {3} {16} - \ varepsilon $
A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\varepsilon$
论文作者
论文摘要
令$ x $为紧凑的连接双曲表面，即，具有恒定曲率-1的riemannian度量的封闭连接的定向平滑表面。对于\ mathbf {n} $中的每个$ n \，令$ x_ {n} $为一个随机度 - $ n $ cover，$ x $ cover的$ x $均匀地采样 - $ n $ n $ riemannian覆盖$ x $的空间。 $ x $或$ x_ {n} $的特征值是关联的laplacian操作员$δ_{x} $或$δ__{x_ {n}} $的特征值。我们说，如果出现$ x_n $的特征值是新事物的新事物，那么多重性比$ x $更大。我们证明，对于任何$ \ varepsilon> 0 $，概率趋于1 $ n \ to \ infty $，在$ x_n $下面没有$ x_n $以下$ x_n $ to $ \ frac {3} {16} {16} {16} - \ varepsilon $。我们猜想相同的结果与$ \ frac {3} {16} $由$ \ frac {1} {4} $代替。
Let $X$ be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each $n\in\mathbf{N}$, let $X_{n}$ be a random degree-$n$ cover of $X$ sampled uniformly from all degree-$n$ Riemannian covering spaces of $X$. An eigenvalue of $X$ or $X_{n}$ is an eigenvalue of the associated Laplacian operator $Δ_{X}$ or $Δ_{X_{n}}$. We say that an eigenvalue of $X_n$ is new if it occurs with greater multiplicity than in $X$. We prove that for any $\varepsilon>0$, with probability tending to 1 as $n\to\infty$, there are no new eigenvalues of $X_n$ below $\frac{3}{16}-\varepsilon$. We conjecture that the same result holds with $\frac{3}{16}$ replaced by $\frac{1}{4}$.
