The Fourier spectrum of critical percolation
The Fourier spectrum of critical percolation
复制标题
临界渗流的傅立叶谱
DOI：
10.1007/s11511-010-0051-x
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发表时间：
2008
期刊：
影响因子：
3.7
通讯作者：
O. Schramm
中科院分区：
文献类型：
--
作者：
C. Garban;G. Pete;O. Schramm
考虑二维渗透交叉事件的指示函数f。本文研究了f的傅里叶变换，并在几种情况下得到了其下尾的尖锐边界。推导了这些界限的各种应用。特别地，我们证明了在原点渗流的三角网格上，动态临界点渗流的异常时间集几乎肯定具有维数${\frac{31}{36}}$，而在半平面上对应的维数为${\frac{5}{9}}$。同时也证明了临界键渗透在方形网格上几乎肯定存在异常次数。此外，还确定了需要重新采样的站点数量的渐近性，以便在一个大的正方形中显著地扰动全局渗透配置。
Consider the indicator function f of a 2-dimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension ${\frac{31}{36}}$ almost surely, and the corresponding dimension in the half-plane is ${\frac{5}{9}}$ . It is also proved that critical bond percolation on the square grid has exceptional times almost surely. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.