Variance decay for functionals of the environment viewed by the particle

TL;DR

This paper proves polynomial decay of variance for the environment viewed by a particle in a random conductance model, using Nash inequalities and martingale techniques, with implications for the walk's mean square displacement convergence.

Contribution

It establishes the polynomial decay rate of variance in the environment viewed by the particle under bounded conductances, introducing a novel analytical approach.

Findings

Variance decays polynomially fast in the environment viewed by the particle

Nash inequality is key to the analysis

Results imply convergence rate for the walk's mean square displacement

Abstract

For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions, our results imply an estimate of the speed of convergence of the mean square displacement of the walk towards its limit.