Percolation of random nodal lines
Percolation of random nodal lines
复制标题
随机节点线的渗透
DOI：
10.1007/s10240-017-0093-0
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
D. Gayet
中科院分区：
文献类型：
--
作者：
V. Beffara;D. Gayet
证明了实平面上实解析函数的自然无限维空间的节域和节线的Russo-Seymour-Welsh渗流定理。更确切地说，设U$U$是R2$\mathbf{R}^{2}$中的光滑连通有界开集，且在U$U$的边界上有两个长度为正的不相交圆弧γ，γ‘$\Gamma’$。证明了存在一个正的常数c$c$，使得对于任何正的比例S$S$，以至少c$c$的概率存在集合{x∈U‘，f(Sx)>；0}${x\in sMash{\bar{U}}，\f(Sx)>；0的连通分支，其中f$f$是与实γ-Fock空间相交的γ空间中的一个随机解析函数.对于足够大的S$S$，零集{x∈U‘，f(Sx)=0}${x\in\sMash{\bar{U}}，\f(Sx)=0\}$也有相同的结论。作为一个重要的中间结果，我们证明了一般定常高斯场的符号渗流可以等价于格点上的关联渗流模型。
We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let U$U$ be a smooth connected bounded open set in R2$\mathbf{R}^{2}$ and γ,γ′$\gamma, \gamma '$ two disjoint arcs of positive length in the boundary of U$U$. We prove that there exists a positive constant c$c$, such that for any positive scale s$s$, with probability at least c$c$ there exists a connected component of the set {x∈U¯,f(sx)>0}$\{x\in \smash{\bar{U}},\ f(sx) > 0\} $ intersecting both γ$\gamma $ and γ′$\gamma '$, where f$f$ is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For s$s$ large enough, the same conclusion holds for the zero set {x∈U¯,f(sx)=0}$\{x\in \smash{\bar{U}},\ f(sx) = 0\} $. As an important intermediate result, we prove that sign percolation for a general stationary Gaussian field can be made equivalent to a correlated percolation model on a lattice.
影响因子：
2.4
作者：
Beliaev D
通讯作者：
Beliaev D