{"title":"Hyperuniformity in regular trees","authors":"Mattias Byléhn","doi":"arxiv-2409.10998","DOIUrl":null,"url":null,"abstract":"We study notions of hyperuniformity for invariant locally square-integrable\npoint processes in regular trees. We show that such point processes are never\ngeometrically hyperuniform, and if the diffraction measure has support in the\ncomplementary series then the process is geometrically hyperfluctuating along\nall subsequences of radii. A definition of spectral hyperuniformity and stealth\nof a point process is given in terms of vanishing of the complementary series\ndiffraction and sub-Poissonian decay of the principal series diffraction near\nthe endpoints of the principal spectrum. Our main contribution is providing\nexamples of stealthy invariant random lattice orbits in trees whose number\nvariance grows strictly slower than the volume along some unbounded sequence of\nradii. These random lattice orbits are constructed from the fundamental groups\nof complete graphs and the Petersen graph.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"50 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10998","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We study notions of hyperuniformity for invariant locally square-integrable
point processes in regular trees. We show that such point processes are never
geometrically hyperuniform, and if the diffraction measure has support in the
complementary series then the process is geometrically hyperfluctuating along
all subsequences of radii. A definition of spectral hyperuniformity and stealth
of a point process is given in terms of vanishing of the complementary series
diffraction and sub-Poissonian decay of the principal series diffraction near
the endpoints of the principal spectrum. Our main contribution is providing
examples of stealthy invariant random lattice orbits in trees whose number
variance grows strictly slower than the volume along some unbounded sequence of
radii. These random lattice orbits are constructed from the fundamental groups
of complete graphs and the Petersen graph.