  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.
