截断高斯自由场节点集的空穴概率
Hole probability for nodal sets of the cut-off Gaussian Free Field
摘要：
设（$Σ$，g）是具有黎曼度量的闭连通曲面.设（$λ$ n）n $\in$ N和（$ψ$ n）n $\in$ N是$Σ$上拉普拉斯算子的特征值的递增序列和相应的L2归一化特征函数的序列.对于每个L \textgreater{} 0，我们考虑$ϕ$ L = 0 \textless{} $λ$ n $\le$ L $ξ$ n $\sqrt$ $λ$ n $ψ$ n，其中$ξ$ n是方差为1的独立同中心高斯。当L $\rightarrow$ $\infty$时，$ϕ$ L收敛。在分布的意义下，将其推广到$Σ$上的高斯自由场。我们首先计算这个域族的协方差函数的渐近行为为L $\rightarrow$ $\infty$。然后，我们使用这个结果来获得概率的渐近性，$ϕ$ L是积极的一个给定的开放的真子集具有光滑的边界。在这样做时，我们还证明了$ϕ$ L的上确界的浓度在1 $\sqrt$ 2 $π$ ln L附近。
英文摘要：
Let ($Σ$, g) be a closed connected surface equipped with a riemannian metric. Let ($λ$ n) n$\in$N and ($ψ$ n) n$\in$N be the increasing sequence of eigenvalues and the sequence of corresponding L 2-normalized eigenfunctions of the laplacian on $Σ$. For each L \textgreater{} 0, we consider $ϕ$ L = 0\textless{}$λ$n$\le$L $ξ$n $\sqrt$ $λ$n $ψ$ n where the $ξ$ n are i.i.d centered gaussians with variance 1. As L $\rightarrow$ $\infty$, $ϕ$ L converges a.s. to the Gaussian Free Field on $Σ$ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L $\rightarrow$ $\infty$. We then use this result to obtain the asymptotics of the probability that $ϕ$ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of $ϕ$ L around 1 $\sqrt$ 2$π$ ln L.
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