TY - JOUR

T1 - The speed of biased random walk among random conductances

AU - Berger, Noam

AU - Gantert, Nina

AU - Nagel, Jan

N1 - Publisher Copyright:
© Association des Publications de l’Institut Henri Poincaré, 2019.

PY - 2019/5

Y1 - 2019/5

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 1.1. 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 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 1.2.

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 1.1. 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 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 1.2.

KW - Effective velocity

KW - Random conductances

KW - Random walk in random environment

KW - Regeneration times

UR - http://www.scopus.com/inward/record.url?scp=85066146832&partnerID=8YFLogxK

U2 - 10.1214/18-AIHP901

DO - 10.1214/18-AIHP901

M3 - Article

AN - SCOPUS:85066146832

SN - 0246-0203

VL - 55

SP - 862

EP - 881

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 2

ER -