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 \(U\) be a smooth connected bounded open set in \(\mathbf{R}^{2}\) and \(\gamma, \gamma '\) two disjoint arcs of positive length in the boundary of \(U\). We prove that there exists a positive constant \(c\), such that for any positive scale \(s\), with probability at least \(c\) there exists a connected component of the set \(\{x\in \smash{\bar{U}},\ f(sx) > 0\} \) intersecting both \(\gamma \) and \(\gamma '\), where \(f\) is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For \(s\) large enough, the same conclusion holds for the zero set \(\{x\in \smash{\bar{U}},\ f(sx) = 0\} \). 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.

中文翻译：

---

随机节点线的渗流

我们证明了与实平面上实际解析函数的自然无限维空间相关的节点域和节点线的Russo-Seymour-Welsh渗流定理。更精确地说，令\（U \）为\（\ mathbf {R} ^ {2} \）和\（\ gamma，\ gamma'\）中边界长度为正的两个不相交的圆弧的光滑连接有界开放集\（U \）。我们证明存在一个正常数\（c \），使得对于任何正尺度\（s \），概率至少为\（c \），存在一个集合\（\ {x \ in \ smash {\ bar {U}}，\ f（sx）> 0 \} \）与\（\ gamma \）和\（\ gamma'\）相交，其中\（f \）是与真正的Bargmann-Fock空间相关的Wiener空间。对于足够大的\（s \），对零集\（\ smash {\ bar {U}}，\ f（sx）= 0 \} \）得出相同的结论。