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The Alexander-Orbach conjecture holds in high dimensions. (English) Zbl 1180.82094

Summary: 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 S. Alexander and R. Orbach [J. Phys. Lett. (Paris) 43, 625–631 (1982)]. En route we calculate the one-arm exponent with respect to the intrinsic distance.

MSC:

82B43 Percolation
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
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