{"title":"Mean Eigenvector Self-Overlap in the Real and Complex Elliptic Ginibre Ensembles at Strong and Weak Non-Hermiticity","authors":"Mark J. Crumpton, Yan V. Fyodorov, Tim R. Würfel","doi":"10.1007/s00023-024-01530-2","DOIUrl":null,"url":null,"abstract":"<div><p>We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in <span>\\(N\\times N\\)</span> non-Hermitian random Gaussian matrices. In a well-known work by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as <span>\\(N\\rightarrow \\infty \\)</span> (Chalker and Mehlig in Phys Rev Lett 81(16):3367–3370, 1998). In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles, which are characterised by correlations between off-diagonal entries controlled by a parameter <span>\\(\\tau \\in [0,1]\\)</span>, with <span>\\(\\tau =1\\)</span> corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite <i>N</i>, for any eigenvalue off the real axis. We further investigate several scaling regimes as <span>\\(N\\rightarrow \\infty \\)</span>, both in the limit of strong non-Hermiticity keeping a fixed <span>\\(\\tau \\in [0,1)\\)</span> and in the weak non-Hermiticity limit, with <span>\\(\\tau \\)</span> approaching unity in such a way that <span>\\(N(1-\\tau )\\)</span> remains finite.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 6","pages":"2069 - 2116"},"PeriodicalIF":1.3000,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00023-024-01530-2.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Henri Poincaré","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1007/s00023-024-01530-2","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in \(N\times N\) non-Hermitian random Gaussian matrices. In a well-known work by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as \(N\rightarrow \infty \) (Chalker and Mehlig in Phys Rev Lett 81(16):3367–3370, 1998). In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles, which are characterised by correlations between off-diagonal entries controlled by a parameter \(\tau \in [0,1]\), with \(\tau =1\) corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite N, for any eigenvalue off the real axis. We further investigate several scaling regimes as \(N\rightarrow \infty \), both in the limit of strong non-Hermiticity keeping a fixed \(\tau \in [0,1)\) and in the weak non-Hermiticity limit, with \(\tau \) approaching unity in such a way that \(N(1-\tau )\) remains finite.
期刊介绍:
The two journals Annales de l''Institut Henri Poincaré, physique théorique and Helvetica Physical Acta merged into a single new journal under the name Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics edited jointly by the Institut Henri Poincaré and by the Swiss Physical Society.
The goal of the journal is to serve the international scientific community in theoretical and mathematical physics by collecting and publishing original research papers meeting the highest professional standards in the field. The emphasis will be on analytical theoretical and mathematical physics in a broad sense.