OAI.DirectionalZeroOne.directional_zero_one

The theorem states that, for every dimension d ≥ 3, a random walk in an i.i.d. random environment on the lattice ℤ^d obeys a directional zero-one law. Here a row at a site is a probability vector p on the 2d nearest-neighbour steps (coordinate direction i in Fin d, sign + or −), with each p(e) in [0,1] and the entries summing to 1. The environment assigns an independent row to every site, each drawn from a probability measure μ on rows, and the strict ellipticity hypothesis on μ says that μ-almost every row gives every one of the 2d steps strictly positive probability. Given the environment, the walker at site x moves to x plus the unit vector of step e with probability p_x(e); the annealed law of the path started at 0 averages over both the environment and the walk. For any nonzero real vector ℓ in ℝ^d, the escape event is that the path X satisfies Σᵢ Xₙ(i)·ℓᵢ → +∞ as n → ∞, that is, the walk tends to infinity in the direction ℓ. The conclusion is that the annealed probability of this event, starting from 0, is either exactly 0 or exactly 1.