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Mean-field critical behaviour for percolation in high dimensions. (English) Zbl 0698.60100
Summary: The triangle condition for percolation states that \(\sum_{x,y}\tau (0,x)\tau (x,y)\tau (y,0)\) is finite at the critical point, where \(\tau\) (x,y) is the probability that the sites x and y 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 the d-dimensional hypercubic lattice, if d 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 \((\gamma =\beta =1\), \(\delta =\Delta_ t=2\), \(t\geq 2)\) and that the percolation density is continuous at the critical point. We also prove that \(\nu_ 2=1/2\) in (i) and (ii), where \(\nu_ 2\) is the critical exponent for the correlation length.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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