Corner percolation on ℤ2 and the square root of 17

Corner percolation on ℤ² and the square root of 17

Gábor Pete

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

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)≈n^−γ and the expectation $\mathbb{E}$ (length of a typical cycle with diameter n)≈n^δ, 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.

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

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