{"title":"Moments of traces of random symplectic matrices and hyperelliptic $L$-functions","authors":"Alexei Entin, Noam Pirani","doi":"arxiv-2409.04844","DOIUrl":null,"url":null,"abstract":"We study matrix integrals of the form\n$$\\int_{\\mathrm{USp(2n)}}\\prod_{j=1}^k\\mathrm{tr}(U^j)^{a_j}\\mathrm d U,$$\nwhere $a_1,\\ldots,a_r$ are natural numbers and integration is with respect to\nthe Haar probability measure. We obtain a compact formula (the number of terms\ndepends only on $\\sum a_j$ and not on $n,k$) for the above integral in the\nnon-Gaussian range $\\sum_{j=1}^kja_j\\le 4n+1$. This extends results of\nDiaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral\nvalid in the (Gaussian) range $\\sum_{j=1}^kja_j\\le n$ and $\\sum_{j=1}^kja_j\\le\n2n+1$ respectively. We derive our formula using the connection between random\nsymplectic matrices and hyperelliptic $L$-functions over finite fields, given\nby an equidistribution result of Katz and Sarnak, and an evaluation of a\ncertain multiple character sum over the function field $\\mathbb F_q(x)$. We\napply our formula to study the linear statistics of eigenvalues of random\nunitary symplectic matrices in a narrow bandwidth sampling regime.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"40 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-07","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-2409.04844","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 matrix integrals of the form
$$\int_{\mathrm{USp(2n)}}\prod_{j=1}^k\mathrm{tr}(U^j)^{a_j}\mathrm d U,$$
where $a_1,\ldots,a_r$ are natural numbers and integration is with respect to
the Haar probability measure. We obtain a compact formula (the number of terms
depends only on $\sum a_j$ and not on $n,k$) for the above integral in the
non-Gaussian range $\sum_{j=1}^kja_j\le 4n+1$. This extends results of
Diaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral
valid in the (Gaussian) range $\sum_{j=1}^kja_j\le n$ and $\sum_{j=1}^kja_j\le
2n+1$ respectively. We derive our formula using the connection between random
symplectic matrices and hyperelliptic $L$-functions over finite fields, given
by an equidistribution result of Katz and Sarnak, and an evaluation of a
certain multiple character sum over the function field $\mathbb F_q(x)$. We
apply our formula to study the linear statistics of eigenvalues of random
unitary symplectic matrices in a narrow bandwidth sampling regime.