Nina Gantert (Munich)
Einstein relation and monotonicity of the speed for random walk among
random conductances
Abstract:
Many applications, such as porous media or composite materials, involve
heterogeneous media which are modeled by random fields. These media are
locally irregular but are "statistically homogeneous" in the sense that
their law has homogeneity properties. Considering random motions in such
a random medium, it turns out often that they can be described by their
effective behaviour. This means that there is a deterministic medium,
the effective medium, whose properties are close to the random
medium, when measured on long space-time scales. In other words, the
local irregularities of the random medium average out over large
space-time scales, and the random motion is characterized by the
"macroscopic" parameters of the effective medium. How do the
macroscopic parameters depend on the law of the random medium?
As an example, we consider the effective diffusivity (i.e. the
covariance matrix in the central limit theorem) of a random walk among
random conductances. It is interesting
and non-trivial to describe this diffusivity in terms of the law of the
conductances. The Einstein relates this diffusivity with the derivative
of the speed of a biased random walk among random conductances. We
explain the Einstein relation and
we also discuss monotonicity questions for the speed of a biased random
walk among random conductances.
The talk is based on joint work (in progress) with Noam Berger, Xiaoqin
Guo and and Jan Nagel.