The Annals of Probability

Corner percolation on ℤ2 and the square root of 17

Gábor Pete

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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.

Article information

Source
Ann. Probab., Volume 36, Number 5 (2008), 1711-1747.

Dates
First available in Project Euclid: 11 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1221138764

Digital Object Identifier
doi:10.1214/07-AOP373

Mathematical Reviews number (MathSciNet)
MR2440921

Zentralblatt MATH identifier
1159.60032

Keywords
Dependent percolation dimer models critical exponents additive Brownian motion simple random walk excursions conformal invariance

Citation

Pete, Gábor. Corner percolation on ℤ 2 and the square root of 17. Ann. Probab. 36 (2008), no. 5, 1711--1747. doi:10.1214/07-AOP373. https://projecteuclid.org/euclid.aop/1221138764


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