The speed of biased random walk among random conductances

Abstract

We consider biased random walk among iid, uniformly elliptic conductances on Z d , and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 3: it follows along the lines of the proof of the Einstein relation in [GGN]. On the other hand, we give a counterexample showing that for iid, uniformly elliptic conductances, the velocity is not always increasing as a function of the bias. More precisely, if d = 2 and if the conductances take the values 1 (with probability p) and κ (with probability 1 − p) and p is close enough to 1 and κ small enough, the velocity is not increasing as a function of the bias, see Theorem 2.

Regularity of biased 1D random walks in random environment

“…As a byproduct of (54), (55) and (56) we conclude that E[Z 0 · · · Z n ] ≤ n 1−γ+o (1) . The above result and (53) imply our claim (52).…”

“…One of the most interesting and most studied models is that of a walk on the infinite supercritical percolation cluster, where the speed has been proved to be positive up to a critical value of λ and equal to zero above this threshold [3,13]. This non-monotone nature of the speed as a function of the bias has been also recently observed for walks among elliptic conductances [1]. Results concerning the continuity of the speed have been obtained e.g.…”

“…In fact, on the one hand it holds E[Z 0 · · · Z n ] ≥ E[Z 1 · · · Z n 1 τ 1 >n ]/2 = P(τ 1 > n)/2 ≥ (c 1 /2)(n + 1) 1−γ = n 1−γ+o (1) . (53) On the other hand, we take A(n) := (γ − 1) log 2 n and we write…”

“…The solid line represents the function σ 2 (λ) in the case of i.i.d. conductances sampled from a uniform random variable in the interval[1,10]. This is plotted against the function σ 2 (λ) (dotted line) in the deterministic case of constant conductances, cfr.…”

The speed of random walk on Galton-Watson trees with vanishing conductances

“…The monotonicity of v as a function of λ has been studied by [5], the behavior for λ close to the recurrent regime by [6] and differentiability by [11]. Closely related are results for random walks in Z d with random conductances and bias parameter λ, as studied by [17,18,7], or for the Mott random walk [15,16]. The regularity of the speed on the tree as a function of the offspring law was studied in [20], when the offspring law is close to criticality.…”