{"title":"Spectral moments of the real Ginibre ensemble","authors":"Sung-Soo Byun, Peter J. Forrester","doi":"10.1007/s11139-024-00879-6","DOIUrl":null,"url":null,"abstract":"<p>The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large <i>N</i> expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments <span>\\(M_{2p}^\\textrm{r}\\)</span>. The latter are expressed in terms of the <span>\\({}_3 F_2\\)</span> hypergeometric functions, with a simplification to the <span>\\({}_2 F_1\\)</span> hypergeometric function possible for <span>\\(p=0\\)</span> and <span>\\(p=1\\)</span>, allowing for the large <i>N</i> expansion of these moments to be obtained. The large <i>N</i> expansion involves both integer and half-integer powers of 1/<i>N</i>. The three-term recurrence then provides the large <i>N</i> expansion of the full sequence <span>\\(\\{ M_{2p}^\\textrm{r} \\}_{p=0}^\\infty \\)</span>. Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large <i>N</i> expansion of these quantities are determined.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"219 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00879-6","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
The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments \(M_{2p}^\textrm{r}\). The latter are expressed in terms of the \({}_3 F_2\) hypergeometric functions, with a simplification to the \({}_2 F_1\) hypergeometric function possible for \(p=0\) and \(p=1\), allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence \(\{ M_{2p}^\textrm{r} \}_{p=0}^\infty \). Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined.