论文标题
随机三型型模型
A model for random three--manifolds
论文作者
论文摘要
我们研究了紧凑的三拟合，并通过将截短的四面体沿其面部随机粘合在一起，从而获得了边界。我们证明，由于四面体的数量倾向于无穷大，这些歧管几乎可以肯定地毫无疑问，这些歧管已连接并具有单个边界成分。我们证明了该边界成分属的大量定律，我们表明，这些歧管的Heegaard属在四面体的数量中是线性的，我们绑定了他们的第一个Betti数字。 我们还表明，几乎肯定是因为四面体的数量倾向于无穷大，我们的歧管承认了一个独特的双曲线指标，具有完全地球的边界。我们证明了该度量量的大量定律，证明了相关的拉普拉斯人具有均匀的光谱间隙，并表明我们的歧管的直径与其体积的关系是对数。最后，我们确定了我们随机歧管序列的本杰明 - schramm极限。
We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component. We prove a law of large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number. We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary. We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini--Schramm limit of our sequence of random manifolds.
