Electronic Journal of Probability

Functional CLT for Random Walk Among Bounded Random Conductances

Marek Biskup and Timothy Prescott

Full-text: Open access

Abstract

We consider the nearest-neighbor simple random walk on $Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $\omega$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk returns to the starting point at a given time.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 49, 1323-1348.

Dates
Accepted: 25 October 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818520

Digital Object Identifier
doi:10.1214/EJP.v12-456

Mathematical Reviews number (MathSciNet)
MR2354160

Zentralblatt MATH identifier
1127.60093

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60F05: Central limit and other weak theorems 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Random conductance model invariance principle corrector homogenization heat kernel percolation isoperimetry

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Biskup, Marek; Prescott, Timothy. Functional CLT for Random Walk Among Bounded Random Conductances. Electron. J. Probab. 12 (2007), paper no. 49, 1323--1348. doi:10.1214/EJP.v12-456. https://projecteuclid.org/euclid.ejp/1464818520


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