Journal article
The Alexander-Orbach conjecture holds in high dimensions
Inventiones Mathematicae, Vol.178(3), pp.635-654
Dec/2009
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 (J. Phys. Lett. (Paris) 43:625-631, 1982). En route we calculate the one-arm exponent with respect to the intrinsic distance.
Details
- Title
- The Alexander-Orbach conjecture holds in high dimensions
- Creators
- Gady Kozma (null) - 972WIS_INST___84Asaf Nachmias (null)
- Resource Type
- Journal article
- Publication Details
- Inventiones Mathematicae, Vol.178(3), pp.635-654; Dec/2009
- Number of pages
- 20
- Language
- English
- DOI
- https://doi.org/10.1007/s00222-009-0208-4
- Grant note
- NSF [DMS-0605166]The research of AN was supported in part by NSF grant # DMS-0605166. Part of this work was carried out while G. K. was a visitor at IMPA and A. N. was a visitor at the Theory Group of Microsoft Research. We would like to extend our gratefulness for the kind hospitality of both institutions, and especially that of Vladas Sidoravicius._ALMAME_DELIMITER_
- Record Identifier
- 993267752903596
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