On slowdown and speedup of transient random walks in random environment

Abstract

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^\kappa)$ from the origin, $\kappa\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{\nu_0})$ from the origin, $\nu_0\in (0,\kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{\nu_1}$ from the origin, $\nu_1\in (\kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^\nu)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.Comment: 43 pages, 4 figures; to appear in Probability Theory and Related Field

Maximal displacement for bridges of random walks in a random environment

“…Proof. This lemma follows easily from Lemma 2.3 and the moderate deviation asymptotics derived in [FGP10]. If ν ∈ (0, 1 ∧ κ), then Theorem 1.2 in [FGP10] implies that lim n→∞ ln (− ln P ω (T n ν > n)) ln n = lim n→∞ ln (− ln P ω (X n < n ν )) ln n…”

Random walks in a sparse random environment

“…Furthermore, the maximum is attained at P ∈ P • µ,ν if and only if VAR P (d 0 ) = VAR P (λ 0 ) = 0, in which case λ 0 = 1 1+e b , P − a. s. The slowdown of a one-dimensional random walk in a random environment, as compared to a simple random walk, is a well-known general phenomenon [11,32,37] that can be explained heuristically by fluctuations in the associated random potential. For example, a random walk transient to the right will quickly pass stretches of the environment that "push" it forward, but will be "trapped" for a long time in atypical stretches that "push" it backward.…”

“…Furthermore, a careful inspection of the proof given in Section 4.3 shows that both parameters b and b of the limiting distributions are decreasing functions of E P (d 0 ) and increasing functions of V AR(d n ). This can be explained by the fact that b, in some rigorous sense, plays the role of the variance for the stable laws L κ,b ; see, for instance, the form of the characteristic function in (11) and compare it to the characteristic function of a normal distribution.…”

Asymptotic direction for random walks in mixing random environments

“…On the other hand, using the sharp estimate in Theorem 1.3 in [FGP10] and denoting byP the law of underlying one-dimensional random walk corresponding to the annealed law of (X n · e 1 ) n≥0 , we can see that for large L, there exists a positive constant K 2 such that…”