The Alexander-Orbach conjecture holds in high dimensions

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.

25 pages, 2 figures. To appear in Inventiones Mathematicae

References in corpus (4)

• Random walks on supercritical percolation clusters
• Critical random graphs: Diameter and mixing time
• Critical percolation in the plane
• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

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