随机平面三角剖分空间与态密度
Spaces of Random Plane Triangulations and the Density of States
摘要：
平铺空间是使用一个度量构造的，其中两个$\mathbb{R}^n$的平铺是接近的当且仅当，经过一个小的平移，它们在原点周围的一个大球上达成一致。我们构造类似的空间来研究平面的随机三角剖分。我们构造了一个连续空间，它是一个具有横截测度的叶状空间，和一个离散空间，它是该空间的一个横截。这些三角剖分上的测度可以构造为球面上测度的极限。我们认为冯诺依曼代数与这些空间。在一定条件下，证明了与离散空间相关联的von Neumann代数是超有限型$II_1$因子.我们还表明，某些运营商的状态密度是行为良好的措施的收敛，特别是可以计算近似球，本征值可以直接计算。此外，我们证明了跳跃的综合态密度和compounds支持的本征函数类似于arXiv：0709.2836之间的连接。
英文摘要：
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.
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