The speed of biased random walk among random conductances

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

Research output: Contribution to journalArticleAcademic

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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 languageEnglish
Article number1704.08844v1
Number of pages24
JournalarXiv
Publication statusPublished - 2017

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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.
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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.

Research output: Contribution to journalArticleAcademic

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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

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