The Alexander-Orbach conjecture holds in high dimensions
By Gady Kozma, Asaf Nachmias. 2008
141 citations recorded by the source corpus.
AbstractWe 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, 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.
Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/322e545847444db366e0266488a58662533986e9
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In Condensed Matter Physics
In Mathematical Physics
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In Statistics and Probability
- Takashi Kumagai
- Marek Tomasz Biskup
- Martin T. Barlow
- Richard F. Bass
- Zhen-Qing Chen
- Alexander Grigorʼyan
- Jiaxin Hu
- M. Kassmann
Citations in the corpus
- Recent progress on the Random Conductance Model (2011, background): “lly speaking, the percolation cluster does not exist but one can still enforce it by conditioning. For the resulting incipient infinite cluster (IIC) in sufficiently high dimensions, Nachmias and Kozma [96] proved the Alexander-Orbach conjecture in all dimensions d 7 — modulo caveats regarding the exi”
- Uniqueness of Brownian motion on Sierpinski carpets (2008, background): “After this work, little progress was made on critical percolation clusters until the recent papers [8, 7, 24].”