Lyapunov exponents of Green’s functions for random potentials tending to zero

Abstract

We consider quenched and annealed Lyapunov exponents for the Green's function of − + γ V , where the potentials V (x), x ∈ Z d , are i.i.d. nonnegative random variables and γ > 0 is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like c √ γ as γ tends to 0. Here the constant c is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.

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

“…It was shown in [KMZ12] that if E[V ] < +∞ (we use E[V ] as shorthand for E[V (0)]), then as λ tends to 0,…”

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

“…Recently, [Wa01a,Wa02,KMZ12] studied this question under the additional assumption that E[V ] is finite (where we write E[V ] as shorthand for E[V (0)]). They found that, as β tends to 0,…”

Crossing Speeds of Random Walks Among “Sparse” or “Spiky” Bernoulli Potentials on Integers

“…Discussion and an open problem. There are many papers concerning the relationship of the quenched and annealed Lyapunov exponents of a random walk in a random potential on Z d , d ≥ 1 (see [Fl08], [Zy09], [KMZ11], [IV12c], [Zy12], and references therein). Lyapunov exponents represent the exponential decay rates of the quenched and annealed survival probabilities, P ω (τ y < ∞) and E(P ω (τ y < ∞)) respectively.…”