Gordon Slade

Mean-Field Critical Behaviour for Percolation in High Dimensions

AbstractThe triangle condition for percolation states that

The lace expansion for self-avoiding walk in five or more dimensions.

\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)}

On the upper critical dimension of lattice trees and lattice animals

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

Mean-Field Behaviour and the Lace Expansion

(\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{

Self-avoiding walk enumeration via the lace expansion

\smash{\scriptscriptstyle-}

The diffusion of self-avoiding random walk in high dimensions

}}{ \geqslant } 2)