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Nonhermitean Random Matrix Models : a Free Random Variable Approach
High Energy Physics - Phenomenology · Submitted 26 Sep 1996
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Using the standard concepts of free random variables, we show that for a large class of nonhermitean random matrix models, the support of the eigenvalue distribution follows from their hermitean analogs using a conformal transformation.
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The abstract, sentence by sentence
Using the standard concepts of free random variables, we show that for a large class of nonhermitean random matrix models, the support of the eigenvalue distribution follows from their hermitean analogs using a conformal transformation. We also extend the concepts of free random variables to the class of nonhermitean matrices, and apply them to the models discussed by Ginibre-Girko (elliptic ensemble) and Mahaux-Weidenm\"uller (chaotic resonance scattering).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Using the standard concepts of free random variables, we show that for a large class of nonhermitean random matrix models, the support of the eigenvalue distribution follows from their hermitean analogs using a conformal transformation. | result 0.85 | 0.94 | 0.80 | asserted 1.00 |
| s1 | We also extend the concepts of free random variables to the class of nonhermitean matrices, and apply them to the models discussed by Ginibre-Girko (elliptic ensemble) and Mahaux-Weidenm\"uller (chaotic resonance scattering). | contribution 0.98 | 0.93 | 0.15 | asserted 1.00 |
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The paper
Original abstract
Using the standard concepts of free random variables, we show that for a large class of nonhermitean random matrix models, the support of the eigenvalue distribution follows from their hermitean analogs using a conformal transformation. We also extend the concepts of free random variables to the class of nonhermitean matrices, and apply them to the models discussed by Ginibre-Girko (elliptic ensemble) and Mahaux-Weidenm\"uller (chaotic resonance scattering).
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References
How the literature uses it
5 arXiv papers cite this in 6 sentences
Papers citing it · 2 sentences without a confident label
Uses its method or tool 2 papers
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(i) “Duplication” [ref] (cf. also [ref] )—the nonholomorphic Green function ([ref]) has a denominator quadratic in $𝐗$, which makes its evaluation hard, and one needs to linearize it by introducing the ($2 ×2$) “matrix-valued Green function,” [equation] where for short, [equation] and the “block–trace,” [equation] In other words, this object resembles the holomorphic Green function ([ref]), and thus can be approached by methods designed for Hermitian random matrices; the cost being the need to work with $2 ×2$ matrices rather than complex numbers.
Generalized Bures products from free probability · Quaternion free probability and model $𝐒$ · 0.94
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This problem has been resolved by “duplication” [ref] (cf. also [ref] for a slightly different approach), i.e., introducing a $2 N ×2 N$ matrix, [equation] where for short, [equation] This form exactly parallels the holomorphic one ([ref]), with a denominator linear in the random matrix, for which the cost is the doubling of the matrix dimensions.
Hermitian and non-Hermitian covariance estimators for multivariate Gaussian and non-Gaussian assets from random matrix theory · Non-Hermitian Green functions · 0.90
Cites it as background 2 papers
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(iii) They contain two operations widely investigated in the literature on random matrices, albeit rather separately: summation (cf. e.g. [ref] ) and multiplication (cf. e.g. [ref] ).
Generalized Bures products from free probability · Interesting mathematical properties · 0.90
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In particular, the problems of summing ( [ref] ) and multiplying ( [ref] ) non–Hermitian random matrices have been drawing considerable attention — and the model $𝐖$ includes both these operations.
Summing free unitary random matrices · Study of non–Hermitian random matrices · 0.87
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