Spaces of Random Plane Triangulations and the Density of States

TL;DR

This paper constructs and analyzes spaces of random plane triangulations, demonstrating that their associated von Neumann algebras are hyperfinite type II_1 factors and establishing methods to compute the density of states through sphere approximations.

Contribution

It introduces new continuous and discrete spaces for random plane triangulations, proves the associated von Neumann algebra is a hyperfinite type II_1 factor, and links density of states jumps to eigenfunctions.

Findings

Von Neumann algebra of the discrete space is hyperfinite type II_1

Density of states can be computed via sphere approximations

Connection between density of states jumps and eigenfunctions

Abstract

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.