We study a variant of the parallel Moser-Tardos Algorithm. We prove that if we restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables. This is achieved by using the same random bits to resample variables which are far enough in the dependency graph. There are two corollaries. First, we obtain a deterministic algorithm for finding a satisfying assignment, which for any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables. Second, we present a Borel version of the Lovász Local Lemma.