Discrete Flat Plane

The 2 dimensional plane can be characterized as the only simply-connected two dimensional flat manifold. In the discrete there is a similar uniqueness. The discrete hexagonal plane is the only 2-manifold that is flat and simply connected. Compact flat 2-manifolds like the torus or Klein bottle are not simply connected and the cylinder is not simply connected. The two dimensional discrete plane in the discrete has a nice fixed point property. It is the only 2-dimensional graph that is self-dual if we understand the dual graph as the soft Barycentric refinement. We talked about this last time. The naive dual graph obtained by taking the facets and connect two if they intersect in a d-1 dimensional face is one dimensional for manifold as there are no triangles. The soft Barycentric refinement is a triangulation of the dual graph. In one dimension we have cyclic graphs and the one dimensional discrete line as fixed points. In dimension d larger than 1, any d-dimensional graph with a pair of adjacent d-simplices has the property that the density of states converges in law to a universal limit that is only d-dependent. Unlike stated in the talk, this is not the case in dimension 1. The first interesting case is dimension 2 and there, the limiting density of states is actually the spectrum of a graph Laplacian, the Hexagonal lattice, the discrete flat plane. In dimensions d larger than 2, the soft Barycentric central limit theorem still works but the limit is not an operator coming from a graph but from an almost periodic operator on some compact topological group. In one dimension for the Barycentric central limit, the limiting density of states was the equilibrium measure on [0,4]. The potential of this equilibrium measure is constant 0 on the interval. Now, in the soft Barycentric case in 2 dimensions, the spectrum is [0,9] and the density of states measure produces a potential that is piece-wise linear on the spectrum (at least this is what we measure). The density of states therefore should be computable as a derivative of a Hilbert transform. So far, I had not been able yet to find explicit expressions for the density of states. The tools of probability theory do not quite work because the multiplication operator in the Fourier picture is and the third part is correlated to the independent random variables so that we can not just take convolutions. Here are numerically computed CDF and PDF from the distribution of the random variable L on the probability space with normalized Haar measure on this torus. The reason for the Van Hove phase transition at the energy value 8 is because the function L(x,y) has saddles which are critical points with crittical values 8. The critical values of L are 0,8,9. With all the hype the last 45 years or so about Fullerene (=2-manifolds with curvature in the range of -1/6,0,1/6} ) I’m a bit surprised that the density of states of the most obvious fullerene, the flat hexagonal triangular plane is not explicitly known. Maybe because graphenes (not two dimensional planes, but trivalent one dimensional graphs) take the first spot of interest given that graphite is made of it. I actually bought a brick of pure graphite for this talk and also found a cool hexagonal talking board decoration. Well, it is soon halloween.