# The speed of biased random walk among random conductances

N. Berger, N. Gantert, J.H. Nagel

### Abstract

We consider biased random walk among iid, uniformly elliptic conductances on Zd, 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 [GGN17]. 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
Original language English 1704.08844v1 24 arXiv Published - 2017

Conductance
Biased
Random walk
Theorem
Albert Einstein
Monotonicity
Counterexample
Disorder
Strictly
Derivative

### Cite this

Berger, N., Gantert, N., & Nagel, J. H. (2017). The speed of biased random walk among random conductances. arXiv, [1704.08844v1 ].
Berger, N. ; Gantert, N. ; Nagel, J.H. / The speed of biased random walk among random conductances. In: arXiv. 2017.
@article{5bf93d37868c4e13ba71101420c3af97,
title = "The speed of biased random walk among random conductances",
abstract = "We consider biased random walk among iid, uniformly elliptic conductances on Zd, 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 [GGN17]. 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",
author = "N. Berger and N. Gantert and J.H. Nagel",
year = "2017",
language = "English",
journal = "arXiv",
publisher = "Cornell University Library",

}

Berger, N, Gantert, N & Nagel, JH 2017, 'The speed of biased random walk among random conductances', arXiv.

The speed of biased random walk among random conductances. / Berger, N.; Gantert, N.; Nagel, J.H.

In: arXiv, 2017.

TY - JOUR

T1 - The speed of biased random walk among random conductances

AU - Berger, N.

AU - Gantert, N.

AU - Nagel, J.H.

PY - 2017

Y1 - 2017

N2 - We consider biased random walk among iid, uniformly elliptic conductances on Zd, 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 [GGN17]. 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

AB - We consider biased random walk among iid, uniformly elliptic conductances on Zd, 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 [GGN17]. 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

UR - https://arxiv.org/abs/1704.08844

M3 - Article

JO - arXiv

JF - arXiv

M1 - 1704.08844v1

ER -

Berger N, Gantert N, Nagel JH. The speed of biased random walk among random conductances. arXiv. 2017. 1704.08844v1 .