A phase transition and stochastic domination in Pippenger’s probabilistic failure model for Boolean networks with unreliable gates

We study Pippenger’s model of Boolean networks with unreliable gates. In this model, the conditional probability that a particular gate fails, given the failure status of any subset of gates preceding it in the network, is bounded from above by some ε. We show that if a Boolean network with n gates is selected at random according to the Barak-Erdős model of a random acyclic digraph, such that the expected edge density is cn logn, and if ε is equal to a certain function of the size of the largest reflexive, transitive closure of a vertex (with respect to a particular realization of the random digraph), then Pippenger’s model exhibits a phase transition at c = 1. Namely, with probability 1−o(1) as n → ∞, we have the following: for 0 ≤ c ≤ 1, the minimum of the probability that no gate has failed, taken over all probability distributions of gate failures consistent with Pippenger’s model, is equal to o(1), whereas for c > 1 it is equal to exp (