Corner percolation on ℤ2 and the square root of 17
Corner percolation on ℤ2 and the square root of 17
复制标题
ℤ2 和 17 的平方根的角渗滤
DOI：
10.1214/07-aop373
复制
发表时间：
2005
影响因子：
2.3
通讯作者：
G. Pete
中科院分区：
文献类型：
--
作者：
G. Pete
我们考虑一个由Balint Toth引入的四顶点模型：一个在π 2上的依赖键渗流模型，其中每个边都以1/2的概率存在，每个顶点正好有两个相互垂直的入射边。我们证明了所有的组件是有限的循环几乎肯定，但期望直径的循环包含的起源是无限的。此外，我们还得到了如下临界指数：尾概率P（原点圈的直径>n）<$n -γ和期望E（直径为n的典型圈的长度）<$n δ，其中y =（5 -<$17）/4 = 0.219.δ =（φ 17 + 1）/4 = 1.28.δ的值来自奇异的六阶常微分方程，而y + δ = 3/2对应于模型中自然高度函数的标度极限是可加布朗运动，其水平集的Hausdorff维数为3/2。我们还包括许多悬而未决的问题，例如，某些线性熵模型的共形不变性。
We consider a four-vertex model introduced by Balint Toth: a dependent bond percolation model on Ζ 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) ≈ n -γ and the expectation E(length of a typical cycle with diameter n) ≈ n δ , with y = (5 - √17)/4 = 0.219... and δ = (√17 + 1)/4 = 1.28.... The value of δ comes from a singular sixth order ODE, while the relation y + 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.