{"title":"Correlation structure and resonant pairs for arithmetic random waves","authors":"Valentina Cammarota , Riccardo W. Maffucci , Domenico Marinucci , Maurizia Rossi","doi":"10.1016/j.spa.2024.104525","DOIUrl":null,"url":null,"abstract":"<div><div>The geometry of Arithmetic Random Waves has been extensively investigated in the last fifteen years, starting from the seminal papers (Rudnick and Wigman, 2008; Oravecz et al., 2008). In this paper we study the correlation structure among different functionals such as nodal length, boundary length of excursion sets, and the number of intersection of nodal sets with deterministic curves in different classes; the amount of correlation depends in a subtle fashion from the values of the thresholds considered and the symmetry properties of the deterministic curves. In particular, we prove the existence of <em>resonant pairs</em> of threshold values where the asymptotic correlation is full, that is, at such values one functional can be perfectly predicted from the other in the high energy limit. We focus mainly on the 2-dimensional case but we discuss some specific extensions to dimension 3.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"181 ","pages":"Article 104525"},"PeriodicalIF":1.1000,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastic Processes and their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304414924002333","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
The geometry of Arithmetic Random Waves has been extensively investigated in the last fifteen years, starting from the seminal papers (Rudnick and Wigman, 2008; Oravecz et al., 2008). In this paper we study the correlation structure among different functionals such as nodal length, boundary length of excursion sets, and the number of intersection of nodal sets with deterministic curves in different classes; the amount of correlation depends in a subtle fashion from the values of the thresholds considered and the symmetry properties of the deterministic curves. In particular, we prove the existence of resonant pairs of threshold values where the asymptotic correlation is full, that is, at such values one functional can be perfectly predicted from the other in the high energy limit. We focus mainly on the 2-dimensional case but we discuss some specific extensions to dimension 3.
在过去的15年里，算术随机波的几何特性得到了广泛的研究，从一些开创性的论文开始(Rudnick和Wigman, 2008；Oravecz et al., 2008)。本文研究了不同函数之间的关联结构，如节点长度、偏移集的边界长度、节点集与不同类别的确定性曲线的交点个数；相关性的数量以一种微妙的方式取决于所考虑的阈值和确定性曲线的对称性。特别地，我们证明了阈值共振对的存在性，其中渐近相关是充分的，即在这样的值上，一个泛函可以在高能极限下完美地预测另一个泛函。我们主要关注二维的情况，但我们讨论了一些特定的扩展到三维。
期刊介绍：
Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests.
Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.
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