Approximations by graphs and emergence of global structures

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wave functions approximate solutions of the Schrödinger equation with energy resealed by the billiard dimension. As an example, we analyze a Sinai billiard and a scattering system obtained by attaching a pair of external leads to it. The results illustrate emergence of global structures in large quantum graphs.