Einstein relation and steady states for the random conductance model

TL;DR

This paper proves the Einstein relation for biased random walks in random conductance models on high-dimensional lattices, establishing a connection between velocity derivative and diffusivity through steady state analysis.

Contribution

It introduces an expansion of steady states as a function of bias for the first time in this context, providing convergence rate estimates for dimensions d≥3.

Findings

Einstein relation holds for d≥3 in the random conductance model.

Steady states can be expanded as a function of bias with convergence rates.

Invariant measures for the environment seen from the particle are established.

Abstract

We consider random walk among iid, uniformly elliptic conductances on $\mathbb Z^d$, and prove the Einstein relation (see Theorem 1). It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle. These invariant measures are often called steady states. The Einstein relation follows at least for $d\ge 3$, from an expansion of the steady states as a function of the bias (see Theorem 2), which can be considered our main result. This expansion is proved for $d\ge 3$. In contrast to [11], we need not only convergence of the steady states, but an estimate on the rate of convergence (see Theorem 4).