Communications in Mathematical Physics

, Volume 128, Issue 2, pp 333–391

Mean-field critical behaviour for percolation in high dimensions

  • Takashi Hara
  • Gordon Slade

DOI: 10.1007/BF02108785

Cite this article as:
Hara, T. & Slade, G. Commun.Math. Phys. (1990) 128: 333. doi:10.1007/BF02108785


The triangle condition for percolation states that\(\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} \) is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values\((\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)\) and that the percolation density is continuous at the critical point. We also prove thatv2 in (i) and (ii), wherev2 is the critical exponent for the correlation length.

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Takashi Hara
    • 1
  • Gordon Slade
    • 2
  1. 1.Courant Institute of Mathematical SciencesN.Y.U.New YorkUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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