标题： 随机图、扩展族和非紧双曲曲面的构造具有统一谱间隙 显示英文标题

标题： Random graphs, expanding families and the construction of noncompact hyperbolic surfaces with uniform spectral gaps

摘要： 在本文中，我们引入并分析了一个随机图模型$\mathcal{F}_{\chi,n}$，该模型是由内部顶点和边界顶点组成的配置模型。我们研究了在$\chi$和$n$各种增长条件下，$\mathcal{F}_{\chi,n}$中图的特征值的渐近行为。当$n = o\left(\chi^{\frac{2}{3}}\right)$时，我们证明了该模型中的几乎所有图都是连通的，并构成一个扩展器族。我们还建立了第一个斯泰克洛夫特征值的上界，并确定了无法构造扩展器的情况。此外，我们在临界 regime$n \asymp g$中显式构造了一个扩展族，并将其用于构建一系列具有均匀正谱间隙的完备非紧双曲曲面。 显示英文摘要

摘要： In this paper, we introduce and analyze a random graph model $\mathcal{F}_{\chi,n}$, which is a configuration model consisting of interior and boundary vertices. We investigate the asymptotic behavior of eigenvalues for graphs in $\mathcal{F}_{\chi,n}$ under various growth regimes of $\chi$ and $n$. When $n = o\left(\chi^{\frac{2}{3}}\right)$, we prove that almost every graph in the model is connected and forms an expander family. We also establish upper bounds for the first Steklov eigenvalue, identifying scenarios in which expanders cannot be constructed. Furthermore, we explicitly construct an expanding family in the critical regime $n \asymp g$, and apply it to build a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.