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Critical behavior and the limit distribution for long-range oriented percolation. I

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We consider oriented percolation on \({\mathbb{Z}}^d\times{\mathbb{Z}}_+\) whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on \({\mathbb{Z}}^d\) . Suppose that D(x) decays as |x|dα for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension \(d_c=2(\alpha\wedge2)\). We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to \(e^{-c|k|^{\alpha\wedge2}}\) for some c > 0.

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Correspondence to Akira Sakai.

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Chen, L., Sakai, A. Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Relat. Fields 142, 151–188 (2008).

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  • Critical Exponent
  • Limit Distribution
  • Critical Behavior
  • Contact Process
  • Percolation Parameter