Nodal length fluctuations for arithmetic random waves

• 成果类型: Article
• 署名作者: Krishnapur, Manjunath; Kurlberg, Par; Wigman, Igor
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2013.177.2.8
• 发表日期: 2013
• 页码: 699-737
• 关键词: lattice points; zeta-function; eigenfunctions; sets; domains; lines; kind

摘要

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (arithmetic random waves). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.