## The Annals of Probability

### Einstein relation and steady states for the random conductance model

#### Abstract

We consider random walk among i.i.d., 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 $d\geq3$, 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 $d\geq3$. In contrast to Guo [Ann. Probab. 44 (2016) 324–359], we need not only convergence of the steady states, but an estimate on the rate of convergence (see Theorem 4).

#### Article information

Source
Ann. Probab., Volume 45, Number 4 (2017), 2533-2567.

Dates
Revised: April 2016
First available in Project Euclid: 11 August 2017

https://projecteuclid.org/euclid.aop/1502438434

Digital Object Identifier
doi:10.1214/16-AOP1119

Mathematical Reviews number (MathSciNet)
MR3693969

Zentralblatt MATH identifier
1385.60061

#### Citation

Gantert, Nina; Guo, Xiaoqin; Nagel, Jan. Einstein relation and steady states for the random conductance model. Ann. Probab. 45 (2017), no. 4, 2533--2567. doi:10.1214/16-AOP1119. https://projecteuclid.org/euclid.aop/1502438434

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