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The Alexander-Orbach conjecture holds in high dimensions

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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 \(d_{s}=\frac{4}{3}\) , that is, p t (x,x)=t −2/3+o(1). This establishes a conjecture of Alexander and Orbach (J. Phys. Lett. (Paris) 43:625–631, 1982). En route we calculate the one-arm exponent with respect to the intrinsic distance.

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Authors and Affiliations

  1. The Weizmann Institute of Science, Rehovot, POB 76100, Israel

    Gady Kozma

  2. Microsoft Research, One Microsoft way, Redmond, WA, 98052-6399, USA

    Asaf Nachmias

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  1. Gady Kozma
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  2. Asaf Nachmias
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Correspondence to Asaf Nachmias.

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Kozma, G., Nachmias, A. The Alexander-Orbach conjecture holds in high dimensions. Invent. math. 178, 635–654 (2009). https://doi.org/10.1007/s00222-009-0208-4

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