Random colourings of aperiodic graphs: Ergodic and spectral properties

We study randomly coloured graphs embedded into Euclidean space, whose vertex sets are infinite, uniformly discrete subsets of finite local complexity. We construct the appropriate ergodic dynamical systems, explicitly characterise ergodic measures, and prove an ergodic theorem. For covariant operators of finite range defined on those graphs, we show the existence and self-averaging of the integrated density of states, as well as the non-randomness of the spectrum. Our main result establishes Lifshits tails at the lower spectral edge of the graph Laplacian on bond percolation subgraphs, for sufficiently small probabilities. Among other assumptions, its proof requires exponential decay of the cluster-size distribution for percolation on rather general graphs.