The speed of biased random walk among random conductances

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. 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 $\kappa$ (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 2.