Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points
Probability
2012-02-27 v1
Abstract
We study the pioneer points of the simple random walk on the uniform infinite planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of Angel to the quadrangulation case. Our main result is that, up to polylogarithmic factors, pioneer points have been discovered before the walk exits the ball of radius in the UIPQ. As a result we verify the KPZ relation in the particular case of the pioneer exponent and prove that the walk is subdiffusive with exponent less than 1/3. Along the way, new geometric controls on the UIPQ are established.
Cite
@article{arxiv.1202.5454,
title = {Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points},
author = {Itai Benjamini and Nicolas Curien},
journal= {arXiv preprint arXiv:1202.5454},
year = {2012}
}