Random substitution tilings and deviation phenomena

Suppose a set of prototiles allows N different substitution rules. In this paper we study tilings of Rd constructed from random application of the substitution rules. The space of all possible tilings obtained from all possible combinations of these substitutions is the union of all possible tilings spaces coming from these substitutions and has the structure of a Cantor set. The renormalization cocycle on the cohomology bundle over this space determines the statistical properties of the tilings through its Lyapunov spectrum by controlling the deviation of ergodic averages of the Rd action on the tiling spaces.