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 $j\ge 1$, 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.

Published in at http://dx.doi.org/10.1214/11-AOP740 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• An introduction to the dimer model
• Exact bond percolation thresholds in two dimensions
• Bond percolation on isoradial graphs: criticality and universality

Cited by in corpus (8)

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• Bond percolation on isoradial graphs: criticality and universality
• Universality for the random-cluster model on isoradial graphs
• The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
• The $Z$-invariant Ising model via dimers
• Rotational invariance in critical planar lattice models
• Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions
• Three theorems in discrete random geometry