Open Access
July 2013 Inhomogeneous bond percolation on square, triangular and hexagonal lattices
Geoffrey R. Grimmett, Ioan Manolescu
Ann. Probab. 41(4): 2990-3025 (July 2013). DOI: 10.1214/11-AOP729


The star–triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular and hexagonal lattices. Among the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step toward proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.


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Geoffrey R. Grimmett. Ioan Manolescu. "Inhomogeneous bond percolation on square, triangular and hexagonal lattices." Ann. Probab. 41 (4) 2990 - 3025, July 2013.


Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1284.60167
MathSciNet: MR3112936
Digital Object Identifier: 10.1214/11-AOP729

Primary: 60K35
Secondary: 82B43

Keywords: bond percolation , box-crossing , Criticality , Inhomogeneous percolation , RSW lemma , star–triangle transformation , Universality , Yang–Baxter equation

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 4 • July 2013
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