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September 2008 Corner percolation on ℤ2 and the square root of 17
Gábor Pete
Ann. Probab. 36(5): 1711-1747 (September 2008). DOI: 10.1214/07-AOP373

Abstract

We consider a four-vertex model introduced by Bálint Tóth: a dependent bond percolation model on ℤ2 in which every edge is present with probability 1/2 and each vertex has exactly two incident edges, perpendicular to each other. We prove that all components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. Moreover, we derive the following critical exponents: the tail probability ℙ(diameter of the cycle of the origin >n)≈nγ and the expectation $\mathbb{E}$ (length of a typical cycle with diameter n)≈nδ, with $\gamma=(5-\sqrt{17})/4=0.219\ldots$ and $\delta=(\sqrt{17}+1)/4=1.28\ldots$. The value of δ comes from a singular sixth order ODE, while the relation γ+δ=3/2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive Brownian motion, whose level sets have Hausdorff dimension 3/2. We also include many open problems, for example, on the conformal invariance of certain linear entropy models.

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Gábor Pete. "Corner percolation on ℤ2 and the square root of 17." Ann. Probab. 36 (5) 1711 - 1747, September 2008. https://doi.org/10.1214/07-AOP373

Information

Published: September 2008
First available in Project Euclid: 11 September 2008

zbMATH: 1159.60032
MathSciNet: MR2440921
Digital Object Identifier: 10.1214/07-AOP373

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 5 • September 2008
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