TY - JOUR
T1 - The Alexander-orbach conjecture holds in high dimensions
AU - Kozma, Gady
AU - Nachmias, Asaf
PY - 2009/10
Y1 - 2009/10
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=70350608418&partnerID=8YFLogxK
U2 - 10.1007/s00222-009-0208-4
DO - 10.1007/s00222-009-0208-4
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AN - SCOPUS:70350608418
SN - 0020-9910
VL - 178
SP - 635
EP - 654
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -