Translator Disclaimer
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

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.

Citation

Download Citation

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

Information

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

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

Subjects:
Primary: 60K35
Secondary: 82B43

Rights: Copyright © 2013 Institute of Mathematical Statistics

JOURNAL ARTICLE
23 PAGES


SHARE
Vol.41 • No. 5 • September 2013
Back to Top