Percolation Critical Exponents Under the Triangle Condition

Abstract

For independent percolation models, it is shown that if the diagrammatic "triangle condition" is satisfied, then the critical exponents $\delta$ and $\hat\beta$ exist and take their mean-field values, generalizing the criterion introduced in 1984 by Aizenman and Newman for the mean-field value of $\gamma$ in nonoriented percolation. The results apply to a broad class of nonoriented, as well as oriented, weakly homogeneous models, in which the range of the connecting bonds need not be bounded. For the nonoriented case, the condition reduces to the finiteness at the critical point of $\nabla \equiv \sum_{x,y}\tau(0,x)\tau(x,y)\tau(y,0)$ [with $\tau(u,\nu)$ the probability that the site $u$ is connected to $\nu$], which was recently established by Hara and Slade for models with sufficiently spread out connections in $d > 6$ dimensions. Our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter $M(\beta,h)$--whose value at $h = 0+$ yields the percolation density, with $\beta$ parametrizing the bond, or site, occupation probabilities and with $h, h \geq 0$, a "ghost field." The conclusion is that under the triangle condition, in the vicinity of the critical point $(\beta_c,0), M(\beta,0+) \approx (\beta - \beta_c)^{\hat\beta}_+$ and $M(\beta_c,h) \approx h^{1/\delta}$, with $\hat \beta = 1$ and $\delta = 2$.