Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound

Abstract

We consider the simple random walk on Z d , d 3, evolving in a potential of the form βV , where (V (x)) x∈Z d are i.i.d. random variables taking values in [0, +∞), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian −△ + βV .MSC 2010: 82B44, 82D30, 60K37.Keywords: Lyapunov exponents, random walk in random potential, Anderson model.so thatIt is easy to see that f is concave increasing and that f (z) ∼ z as z tends to 0. As a consequence, for any M > 0, (1.10) z M f (βz) dµ(z) ∼ β E[V 1 V M ], while, since f (z) z, z>M f (βz) dµ(z) β E[V 1 V >M ].

Lyapunov exponents of random walks in small random potential: the upper bound

“…We divide up the intervals into two classes, as in [MM12]. We say that the interval I j is relevant if…”

“…This is stochastically dominated by a Binomial random variable with parameters (L λ δ 1 +1) d and P [V (0) ∈ I j ]. By elementary bounds on Binomial tails (see for instance [MM12, (2.16)]), we have for λ small enough,…”

“…We refer to [MM12] for a detailed review of previous results, for motivations and for a heuristic explanation of this formula. We define f : R + → R by…”

“…We refer to [MM12] for a detailed review of previous results, for motivations and for a heuristic explanation of this formula. We define f : R + → R by (1.4) f (z) = q d 1 − e −z 1 − (1 − q d )e −z , so that the integral appearing in the right-hand side of (1.3) can be rewritten as (1.5) I λ = f (λz) dµ(z).…”

“…Since the function f is concave and f (z) ∼ z as z tends to 0, this is consistent with Theorem 1.1. For E[V ] = +∞ and d 3, it was shown in [MM12] that…”