## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 4 (1991), 1520-1536.

### 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 $\delta = 2$.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 4 (1991), 1520-1536.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990221

**Digital Object Identifier**

doi:10.1214/aop/1176990221

**Mathematical Reviews number (MathSciNet)**

MR1127713

**Zentralblatt MATH identifier**

0747.60093

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82A43 34A40: Differential inequalities [See also 26D20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82A25

**Keywords**

Percolation critical exponents triangle diagram differential inequality mean-field behavior power laws

#### Citation

Barsky, D. J.; Aizenman, M. Percolation Critical Exponents Under the Triangle Condition. Ann. Probab. 19 (1991), no. 4, 1520--1536. doi:10.1214/aop/1176990221. https://projecteuclid.org/euclid.aop/1176990221