The Alexander-Orbach conjecture holds in high dimensions
Probability
2015-05-13 v3 Mathematical Physics
math.MP
Abstract
We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.
Cite
@article{arxiv.0806.1442,
title = {The Alexander-Orbach conjecture holds in high dimensions},
author = {Gady Kozma and Asaf Nachmias},
journal= {arXiv preprint arXiv:0806.1442},
year = {2015}
}
Comments
25 pages, 2 figures. To appear in Inventiones Mathematicae