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September 2013 Universality for bond percolation in two dimensions
Geoffrey R. Grimmett, Ioan Manolescu
Ann. Probab. 41(5): 3261-3283 (September 2013). DOI: 10.1214/11-AOP740


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.


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Geoffrey R. Grimmett. Ioan Manolescu. "Universality for bond percolation in two dimensions." Ann. Probab. 41 (5) 3261 - 3283, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1284.60168
MathSciNet: MR3127882
Digital Object Identifier: 10.1214/11-AOP740

Primary: 60K35
Secondary: 82B43

Keywords: arm exponent , bond percolation , box-crossing , Critical exponent , Inhomogeneous percolation , scaling relations , star–triangle transformation , Universality , Yang–Baxter equation

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 5 • September 2013
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