Einstein relation and steady states for the random conductance model

Abstract

We consider random walk among iid, uniformly elliptic conductances on 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 ≥ 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 ≥ 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).

1D Mott Variable-Range Hopping with External Field

“…We point out that a covariance representation of ∂ λ=0 Q λ (f ) as the second one in (12) appears also in [11] and [18].…”

The speed of random walk on Galton-Watson trees with vanishing conductances

“…The monotonicity of v as a function of λ has been studied by [5], the behavior for λ close to the recurrent regime by [6] and differentiability by [11]. Closely related are results for random walks in Z d with random conductances and bias parameter λ, as studied by [17,18,7], or for the Mott random walk [15,16]. The regularity of the speed on the tree as a function of the offspring law was studied in [20], when the offspring law is close to criticality.…”

Random walk on barely supercritical branching random walk

“…In our definition we follow the formulation of [SZ99]. There are two main changes from the classical regeneration time structure of the above mentioned papers: First, we have to allow the random walk to backtrack a distance of order (p − p c ) −1 , similar to the construction in [GMP12,GGN17]. Second, to obtain a stationary sequence even with backtracking, we have to control the environment where the walker regenerates.…”