论文标题
随机平面三角形的空间和状态的密度
Spaces of Random Plane Triangulations and the Density of States
论文作者
论文摘要
使用度量标准构建瓷砖空间，其中两个tilings $ \ mathbb {r}^n $在且只有在小型翻译后才在原点周围的一个大球上达成一致时。我们构建类似的空间来研究平面的随机三角形。我们构建了一个连续空间，该空间是配备横向度量的叶状空间，而离散空间是该空间的横向空间。可以将这些三角形的措施构造为球体措施的限制。 我们考虑与这些空间相关的von Neumann代数。在某些情况下，我们表明与离散空间相关的von Neumann代数是$ ii_1 $ factor的高铁型。我们还表明，某些运算符的状态密度相对于度量的收敛性很好，尤其可以通过在可以直接计算特征值的球体上近似于球体来计算。此外，我们证明了状态集成密度的跳跃与类似于Arxiv的紧凑型征征的跳跃与紧凑的征征：0709.2836。
Tiling spaces are constructed using a metric in which two tilings of $\mathbb{R}^n$ are close if and only if, after a small translation, they agree on a large ball around the origin. We construct analogous spaces to study random triangulations of the plane. We construct a continuous space which is a foliated space equipped with a transverse measure, and a discrete space which is a transversal of that space. Measures on these triangulations can be constructed as limits of measures on spheres. We consider von Neumann algebras associated with these spaces. Under certain conditions, we show that the von Neumann algebra associated with the discrete space is a hyperfinite type $II_1$ factor. We also show that the density of states of certain operators is well-behaved with respect to the convergence of measures, and in particular can be computed by approximating it on spheres, where eigenvalues can be directly computed. Additionally, we prove a connection between jumps of the integrated density of states and compactly supported eigenfunctions analogous to arXiv:0709.2836.
