ar X iv : 0 70 6 . 15 20 v 1 [ m at h . PR ] 1 1 Ju n 20 07 A DYNAMICAL LAW OF LARGE NUMBERS

Let X 1 , X 2 ,. .. denote i.i.d. random bits, each taking the values 1 and 0 with respective probabilities p and 1 − p. A well-known theorem of Erd˝ os and Rényi (1970) describes the length of the longest contiguous stretch, or " run, " of ones in X 1 ,. .. , X n for large values of n. Benjamini, Häggström, Peres, and Steif (2003, Theorem 1.4) demonstrated the existence of unusual times, provided that the bits undergo equilibrium dynamics in time. The first of the two main results of this paper describes what happens if we allow for a fixed and finite number of " impurities " [or zeros] in the longest run of ones. This resolves a recent conjecture of Révész (2005, p. 61). We also compute the Hausdorff dimension of the collection of all unusual times at which this long-run-with-impurities occur. The second main contribution of this paper describes a sharp capacity criterion for a parity test of Benjamini, Häggström, Peres, and Steif (2003) that was initially motivated by problems in complexity theory. This refines the existing sufficient condition and necessary condition of Benjamini, Häggström, Peres, and Steif (2003, Theorem 3.4) to a necessary and sufficient condition which is potential-theoretic in nature. The proof hinges on a combinato-rial argument which does not appear to have an obvious connection to the Markov property. This is worth mentioning because probabilistic potential theory is often associated strongly with the Markov, or even strong Markov, property.