The Alexander-orbach conjecture holds in high dimensions

Gady Kozma, Asaf Nachmias*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

73 Scopus citations


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, that is, pt(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.

Original languageEnglish
Pages (from-to)635-654
Number of pages20
JournalInventiones Mathematicae
Issue number3
StatePublished - Oct 2009
Externally publishedYes


FundersFunder number
National Science Foundation
Directorate for Mathematical and Physical Sciences0605166


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