Introduction and main result Sample Clauses

Introduction and main result. In this chapter we continue to investigate the model in Section 2.1.1. We show that under a certain strong-mixing assumption on the RE ξ, called n-cone-mixing (see Definition 3.1), the RW X satisfies an annealed invariance principle with a Brownian motion as scaling limit. The proof of this functional CLT relies on a direct adaptation of a technique used in [36] for static REs. The n-cone-mixing property is a technical assumption directly connected with the machinery used in the proof. In Section 3.2.4 we will exhibit examples of dynamic REs satisfying this assumption. We first need some definitions. Recall (2.20), and let · denote the Euclidean norm on R2. Put l = (0, 1). For x = (z, m) ∈ H = Z × N0, let CN (x) = {u ∈ R × [0, ∞): √2/2 u − x ≤ (u − x) · l ≤ N } (3.1) be the cone of angle π/2 with tip in (z, m) truncated at time N + m. For fixed L ≥ 0, let {CN (xi): xi = (zi, mi) ∈ H}n be a set of n truncated cones such that, for 1 ≤ i < n, i=1 m1 ≥ L, mi+1 = Ni + mi + L, | zi+1 − zi |≤ Ni. (3.2) We call these nested-cones. In words, we are considering n space-time truncated cones separated in time by a distance L such that the (i + 1)-st cone is contained in the i-th extended cone. .1. Fix L ≥ 0 and any set of n nested-cones {CN (xi): xi = (zi, mi) ∈ H}n . a function Ψ : R+ → R+, with ∫ A dynamic RE ξ on Ω = {0, 1}Z is said to be n-cone-mixing if for any n ∈ N there exists such that ∞ Ψ(t)dt < , (3.3) where A∈Gn,B∈G<n η,η'∈Ω Pη(A | B) − Pη' (A | B) ≤ Ψ(nL), (3.4) Gn = σ{ξs(z): (z, s) ∈ CNn (xn)}, ∪ G<n = σ {ξs(z): (z, s) ∈ n−1 CNi (xi)} .

Introduction and main result. In Section 2.1 we define the random walk in dynamic random environment, introduce a space-time mixing property for the random environment called cone-mixing, and state our law of large numbers for the random walk subject to cone-mixing. In Section 2.2 we give the proof of the law of large numbers with the help of a space-time regeneration- time argument. In Section 2.3 we assume a stronger space-time mixing property, namely, exponential mixing, and derive a series expansion for the global speed of the random walk in powers of the size of the local drifts. This series expansion converges for small enough local drifts and its first term allows us to determine the sign of the global speed. (The perturbation argument underlying the series expansion provides an alternative proof of the law of large numbers.) In Section 2.4 we give examples of random environments that are cone-mixing. In Section 2.5 we compute the first three terms in the expansion for an independent spin-flip dynamics. 2.1.1 Model