Self-similarity under inflation and level statistics: A study in two dimensions

Energy-level spacing statistics are discussed for the octagonal tiling, a two-dimensional quasiperiodic structure. A recursion relation is written for the probability distributions of variables defined on finite-size approximants to this quasiperiodic tiling, using their property of similarity under inflation. Three types of distribution functions are introduced and determined by a combination of numerical and analytical techniques. These are likely to be of general utility in systems lacking translational invariance but with inflation symmetry.