Sharp ellipticity conditions for ballistic behavior of random walks in random environment

Abstract

We sharpen ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ramírez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ramírez. As a corollary, we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension d = 2 under which the corresponding random walk is ballistic.Keywords: Dirichlet distribution; max-flow min-cut theorem; random walk in random environment; reinforced random walks This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2016, Vol. 22, No. 2, 969-994. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2016 ISI/BS 2É. Bouchet, A.F. Ramírez and C. Sabot probabilities P x,ω (X n+1 = x + e|X n = x) = ω(x, e)for each x ∈ Z d and e ∈ U .Let P be a probability measure defined on the environment space Ω endowed with its Borel σ-algebra, such that {ω(x): x ∈ Z d } is i.i.d. under P. We call P x,ω the quenched law of the random walk in random environment (RWRE) starting from x, and P x := P x,ω dP(ω) the averaged or annealed law of the RWRE starting from x. The law P is said to be elliptic if for every x ∈ Z d and e ∈ U , P(ω(x, e) > 0) = 1. We say that P is uniformly elliptic if there exists a constant γ > 0 such that for every x ∈ Z d and e ∈ U , P(ω(x, e) ≥ γ) = 1.Given l ∈ S d−1 we say that the RWRE is transient in direction l if P 0 (A l ) = 1, with A l := lim n→∞ X n · l = ∞ .Furthermore, it is ballistic in direction l if P 0 -a.s. lim inf n→∞

Local trapping for elliptic random walks in random environments in $$\mathbb {Z}^d$$

“…Furthermore, the authors of [6] introduce sufficient ellipticity conditions for ballistic behavior under condition (P) M . Later on, the ellipticity conditions for ballistic behavior were improved in [4], providing an optimal criterion for the case of Dirichlet environments. See Sect.…”

“…In [4], the authors introduced the following condition called (E ) 1 : there exists {φ(e), e ∈ U } ∈ (0, ∞) 2d such that (1) 2 e∈U φ(e) − sup e∈U (φ(e) + φ(−e)) > 1, (2) for every e ∈ U we have that…”

“…Recently, new ellipticity criteria for ballistic behavior have been proved (in [6] and [4]). In this paper, we find a criterion (see Theorem 3.2) for positive speed which extends previously known results.…”

“…In order to illustrate how to apply the criterion (K ) α , we are going to show that (K ) α is more general than the current best criterion for positive speed (see [4]). This will be done in two parts.…”

“…This will be done in two parts. Firstly, we take the criterion for positive speed exhibited in [4] and show that it implies (K 1 ). Secondly, we will provide an example which verifies (K ) 1 but no former criterion.…”