{"title":"Random Hermitian versus random non-Hermitian operators—unexpected links","authors":"A. Jarosz, M. Nowak","doi":"10.1088/0305-4470/39/32/S12","DOIUrl":null,"url":null,"abstract":"We present some unexpected links between the spectral properties of ensembles of large Hermitian and non-Hermitian random matrices, within the formalism of free random variables. We address the task of adding non-Hermitian random matrices. To solve this problem, we present a new approach [1] based on a generalization of the notion of the matrix-valued Green's function to the quaternion level, and on an extension of free probability theory, which allows us to introduce the setting of quaternion-valued free probability theory. We use this quaternion construction to solve the problem of a non-Hermitian random matrix of the type H1 + iH2, with Hermitian H1, H2 freely independent. Finally, we mention conformal mapping relating spectral properties of Hermitian H1 + H2 to spectral properties of the non-Hermitian H1 + iH2 model.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2006-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"30","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of physics A: Mathematical and general","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1088/0305-4470/39/32/S12","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 30
Abstract
We present some unexpected links between the spectral properties of ensembles of large Hermitian and non-Hermitian random matrices, within the formalism of free random variables. We address the task of adding non-Hermitian random matrices. To solve this problem, we present a new approach [1] based on a generalization of the notion of the matrix-valued Green's function to the quaternion level, and on an extension of free probability theory, which allows us to introduce the setting of quaternion-valued free probability theory. We use this quaternion construction to solve the problem of a non-Hermitian random matrix of the type H1 + iH2, with Hermitian H1, H2 freely independent. Finally, we mention conformal mapping relating spectral properties of Hermitian H1 + H2 to spectral properties of the non-Hermitian H1 + iH2 model.