## The Annals of Probability

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

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

Nina Gantert, Xiaoqin Guo, and Jan Nagel

#### 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**

Received: December 2015

Revised: April 2016

First available in Project Euclid: 11 August 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/16-AOP1119

**Mathematical Reviews number (MathSciNet)**

MR3693969

**Zentralblatt MATH identifier**

1385.60061

**Subjects**

Primary: 60K37: Processes in random environments 60K40: Other physical applications of random processes

Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G10: Stationary processes 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

**Keywords**

Random conductance model Einstein relation steady states

#### 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