## The Annals of Probability

### Universality for bond percolation in two dimensions

#### Abstract

All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star–triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent $\rho$, the $2j$-alternating-arms exponents $\rho_{2j}$ for $j\ge1$, the volume exponent $\delta$, and the connectivity exponent $\eta$. By earlier results of Kesten, this implies universality also for the near-critical exponents $\beta$, $\gamma$, $\nu$, $\Delta$ (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.

#### Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3261-3283.

Dates
First available in Project Euclid: 12 September 2013

https://projecteuclid.org/euclid.aop/1378991839

Digital Object Identifier
doi:10.1214/11-AOP740

Mathematical Reviews number (MathSciNet)
MR3127882

Zentralblatt MATH identifier
1284.60168

#### Citation

Grimmett, Geoffrey R.; Manolescu, Ioan. Universality for bond percolation in two dimensions. Ann. Probab. 41 (2013), no. 5, 3261--3283. doi:10.1214/11-AOP740. https://projecteuclid.org/euclid.aop/1378991839

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