The Annals of Probability

Universality for bond percolation in two dimensions

Geoffrey R. Grimmett and Ioan Manolescu

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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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991839

Digital Object Identifier
doi:10.1214/11-AOP740

Mathematical Reviews number (MathSciNet)
MR3127882

Zentralblatt MATH identifier
1284.60168

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

Keywords
Bond percolation inhomogeneous percolation universality critical exponent arm exponent scaling relations box-crossing star–triangle transformation Yang–Baxter equation

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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