The Alexander-Orbach conjecture holds in high dimensions

TL;DR

This paper confirms the Alexander-Orbach conjecture for high-dimensional critical percolation, showing that random walks on the incipient infinite cluster exhibit anomalous diffusion with spectral dimension 4/3.

Contribution

It proves the Alexander-Orbach conjecture in high dimensions and computes the one-arm exponent relative to intrinsic distance.

Findings

Random walk on the IIC shows anomalous diffusion with spectral dimension 4/3.

The spectral dimension d_s=4/3 is established for high-dimensional percolation.

The one-arm exponent with respect to intrinsic distance is calculated.

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.