Meanfield critical behaviour for percolation in high dimensions
Abstract
The triangle condition for percolation states thatsumlimits_{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 sites x and y are connected. We use an expansion related to the lace expansion for a selfavoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearestneighbour independent bond percolation on the ddimensional hypercubic lattice, if d is sufficiently large, and (ii) in more than six dimensions for a class of “spreadout” models of independent bond percolation which are believed to be in the same universality class as the nearestneighbour 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 meanfield (Bethe lattice) values(γ = β = 1,δ = ∆ _t = 2, t\underset{raise0.3emsmashriptscriptstyle}{ ≥slant } 2) and that the percolation density is continuous at the critical point. We also prove that v _{2} in (i) and (ii), where v _{2} is the critical exponent for the correlation length.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 March 1990
 DOI:
 10.1007/BF02108785
 Bibcode:
 1990CMaPh.128..333H