论文标题
单色波的淋巴结长度的小型CLT
Small Scale CLTs for the Nodal Length of Monochromatic Waves
论文作者
论文摘要
我们考虑了限制的节点长度$ l（λ）$ radius $r_λ$ a {\ it高斯回调单色随机波}的参数$λ> 0 $与Riemann Surface $（\ Mathcal M，G）相关的参数$λ> 0 $。我们的主要结果是，如果$r_λ$的生长比$（\logλ）^{1/25} $慢，则（如$λ\ to \ infty $）长度$ l（λ）$验证一个中心极限定理，其与贝里的随机波模型相同，如nourdin，peccati和peccati和sossi（2019）（2019）。利用Bérard（1986）的Keeler（2019）对估计的一些强大扩展，我们的技术主要基于一种新颖的固有限制，以平稳的高斯领域的耦合，这是独立的，这是独立的，并且还可以使我们可以改善nodal长度差异的估算值，以提高nodal rackotic渐变差异的估算。为了证明我们的方法的灵活性，我们还为$ 2 $ torus的缩小球上的算术随机波的淋巴结长度提供了相变的应用。
We consider the nodal length $L(λ)$ of the restriction to a ball of radius $r_λ$ of a {\it Gaussian pullback monochromatic random wave} of parameter $λ>0$ associated with a Riemann surface $(\mathcal M,g)$ without conjugate points. Our main result is that, if $r_λ$ grows slower than $(\log λ)^{1/25}$, then (as $λ\to \infty$) the length $L(λ)$ verifies a Central Limit Theorem with the same scaling as Berry's random wave model -- as established in Nourdin, Peccati and Rossi (2019). Taking advantage of some powerful extensions of an estimate by Bérard (1986) due to Keeler (2019), our techniques are mainly based on a novel intrinsic bound on the coupling of smooth Gaussian fields, that is of independent interest, and moreover allow us to improve some estimates for the nodal length asymptotic variance of pullback random waves in Canzani and Hanin (2016). In order to demonstrate the flexibility of our approach, we also provide an application to phase transitions for the nodal length of arithmetic random waves on shrinking balls of the $2$-torus.
