黎曼随机波的实零点和复零点
Real and complex zeros of Riemannian random waves
摘要：
我们考虑黎曼随机波，即紧致黎曼流形上拉普拉斯算子的本征函数的高斯随机线性组合，其频率来自一个短的间隔（“渐近固定频率”）。我们首先证明了a的实零点集的期望极限分布相对于紧致黎曼流形$(M, g)$的体积形式是一致的。然后，我们表明，复杂的零点集的解析延拓这样的黎曼随机波的Grauert管中的复杂化的$M$趋于极限电流。
英文摘要：
We consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian manifold with frequencies from a short interval (`asymptotically fixed frequency'). We first show that the expected limit distribution of the real zero set of a is uniform with respect to the volume form of a compact Riemannian manifold $(M, g)$. We then show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert tube in the complexification of $M$ tends to a limit current.
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