Abstract
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched uniform invariance principle for the random walk. This means that the rescaled trajectory of length n is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.
Highlights
Introduction and resultsSuppose that for each pair of integers we are given a nonnegative number
The conductances are initially chosen at random, and we call the set of the conductances random environment
One defines a discrete-time random walk in the usual way: the transition probability from x to y is proportional to the conductance between x and y
Summary
Suppose that for each pair of integers we are given a nonnegative number. One may think that sites of Z are nodes of an electrical network where any site can be connected to any other site, and those numbers are thought of as the conductances of the corresponding links. The proof of Theorem 1.1 of [14] relies on the uniform heat-kernel bounds of [13]; one uses these bounds to obtain that, regardless of the starting point, with probability close to 1 the walk will enter to the set of “good” sites (i.e., the sites from where the convergence is good enough) This poses the question of what to do with unbounded conductances (and/or unbounded jumps), to which we have no answer for ( one can expect, as usual, that the case d = 2 should be more accessible, since in this case each site is “surrounded” by “good” sites, cf e.g. the proof of Theorem 4.7 in [5]).
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