Corner percolation on Z2 and the square root of 17

• 成果类型: Article
• 署名作者: Pete, Gabor
• 署名单位: Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP373
• 发表日期: 2008
• 页码: 1711-1747
• 关键词: level sets; dependent percolation; SCALING LIMITS; 6-vertex; CURVES

摘要

We consider a four-vertex model introduced by Balint Toth: a dependent bond percolation model on Z(2) 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) approximate to n(-gamma) and the expection E(length of a typical cycle with diameter n) approximate to n(delta), with gamma = (5 - root 17)/4 = 0.219... and delta = (root 17 + 1)/4 = 0.218... The value of delta comes from a singular sixth order ODE. while the relation gamma + delta = 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.