Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The speed of biased random walk among random conductances

Noam Berger, Nina Gantert, and Jan Nagel

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We consider biased random walk among iid, uniformly elliptic conductances on $\mathbb{Z}^{d}$, 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 1.1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 1.3: it follows along the lines of the proof of the Einstein relation in (Ann. Probab. 45 (4) (2017) 2533–2567). 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 $\kappa$ (with probability $1-p$) and $p$ is close enough to $1$ and $\kappa$ small enough, the velocity is not increasing as a function of the bias, see Theorem 1.2.


Nous étudions des marches aléatoires biaisées dans un milieu aléatoire donné par des poids iid sur les arêtes de $\mathbb{Z}^{d}$. Les poids sont bornés au-dessus et ils ont une borne inférieure qui est strictement positive. Nous nous intéressons pour la vitesse de la marche en fonction du bias. Un argument connu donne que, pour des biais suffisamment grands, la vitesse est une fonction croissante du biais. Notre résultat principal dit que si le désordre est petit, ce qui veut dire que les poids sont proches les uns aux autres, la vitesse est une fonction croissante du bias, voir Théorème 1.1. Un ingrédient crucial de la preuve est une formule pour la dérivée de la vitesse : cette dérivée peut etre écrit comme une covariance, voir Théorème 1.3. La preuve de Théorème 1.3 suis les arguments de la preuve de la relation d’Einstein dans (Ann. Probab. 45 (4) (2017) 2533–2567). Par contre, nous donnons un exemple montrant que pour des poids iid prenant les valeurs $1$ (avec probabilité $p$) et $\kappa$ (avec probabilité $1-p$), si $p$ est suffisamment proche de $1$ et $\kappa$ est suffisamment petit, la vitesse n’est pas une fonction croissante du bias, voir Théorème 1.2.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 862-881.

Received: 27 April 2017
Revised: 13 March 2018
Accepted: 23 March 2018
First available in Project Euclid: 14 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60G42: Martingales with discrete parameter 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Random walk in random environment Random conductances Effective velocity Regeneration times


Berger, Noam; Gantert, Nina; Nagel, Jan. The speed of biased random walk among random conductances. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 862--881. doi:10.1214/18-AIHP901.

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