Abstract
We consider a random walk among unbounded random conductances on the two-dimensional integer lattice. When the distribution of the conductances has an infinite expectation and a polynomial tail, we show that the scaling limit of this process is the fractional kinetics process. This extends the results of the paper [BC10] where a similar limit statement was proved in dimension larger than two. To make this extension possible, we prove several estimates on the Green function of the process killed on exiting large balls.
Highlights
Introduction and main resultsThe main purpose of the present paper is to extend the validity of the quenched non-Gaussian functional limit theorem for random walk among heavy-tailed random conductances on Zd to dimension d = 2
When the distribution of the conductances has an infinite expectation and a polynomial tail, we show that the scaling limit of this process is the fractional kinetics process
This extends the results of the paper [BC10] where a similar limit statement was proved in dimension d ≥ 3
Summary
As in [BC10], we use the fact that the CSRW can be expressed as a time change of another process for which the usual functional limit theorem holds and which can be well controlled This process, called variable speed random walk (VSRW), is a continuoustime Markov chain with transition rates μx y. Green function of the simple random walk, for many centres x and for all y with distance at least r to x and to the boundary of B(x, r) (see Proposition 4.3 in [BC10], cf Lemma 3.5 below) This is shown using a combination of the functional limit theorem for the VSRW and the elliptic Harnack inequality which were both proved in [BD10]. In the appendix we discuss the CSRW among the heavy-tailed conductances on the one-dimensional lattice Z
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