Commutative law for products of infinitely large isotropic random matrices

2013
journal article
article
8
cris.lastimport.wos2024-04-10T00:09:07Z
dc.abstract.enEnsembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size $N\rightarrow \infty$. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example, the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power $A^{n}$ of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as that of $A^{2} B^{2} C^{3}$. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \rightarrow 0$. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit $N \rightarrow \infty$ : The eigenvalue density of an isotropic random matrix has a power-law singularity at the origin $\sim \left | \lambda \right |^{-8}$ with a power $s \epsilon (0,2)$ when and only when the density of its singular values has a power-law singularity $\sim \lambda ^{-\sigma }$ with a power $\sigma$ = s/(4-s). hese results are obtained analytically in the limit $N\rightarrow \infty$. We supplement these results with numerical simulations for large but finite N and discuss finite-size effects for the most common ensembles of isotropic random matrices.pl
dc.affiliationWydział Fizyki, Astronomii i Informatyki Stosowanej : Instytut Fizyki im. Mariana Smoluchowskiegopl
dc.contributor.authorBurda, Zdzisław - 127492 pl
dc.contributor.authorLivan, Giacomopl
dc.contributor.authorŚwięch, Arturpl
dc.date.accessioned2015-06-30T14:18:06Z
dc.date.available2015-06-30T14:18:06Z
dc.date.issued2013pl
dc.description.admin[AB] Święch, Arturpl
dc.description.number2pl
dc.description.publication1pl
dc.description.volume88pl
dc.identifier.articleid022107pl
dc.identifier.doi10.1103/PhysRevE.88.022107pl
dc.identifier.eissn1550-2376pl
dc.identifier.issn1539-3755pl
dc.identifier.urihttp://ruj.uj.edu.pl/xmlui/handle/item/10807
dc.languageengpl
dc.language.containerengpl
dc.rightsDodaję tylko opis bibliograficzny*
dc.rights.licencebez licencji
dc.rights.uri*
dc.subtypeArticlepl
dc.titleCommutative law for products of infinitely large isotropic random matricespl
dc.title.journalPhysical Review. E, Statistical, Nonlinear, and Soft Matter Physicspl
dc.typeJournalArticlepl
dspace.entity.typePublication
cris.lastimport.wos
2024-04-10T00:09:07Z
dc.abstract.enpl
Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size $N\rightarrow \infty$. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example, the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power $A^{n}$ of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as that of $A^{2} B^{2} C^{3}$. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \rightarrow 0$. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit $N \rightarrow \infty$ : The eigenvalue density of an isotropic random matrix has a power-law singularity at the origin $\sim \left | \lambda \right |^{-8}$ with a power $s \epsilon (0,2)$ when and only when the density of its singular values has a power-law singularity $\sim \lambda ^{-\sigma }$ with a power $\sigma$ = s/(4-s). hese results are obtained analytically in the limit $N\rightarrow \infty$. We supplement these results with numerical simulations for large but finite N and discuss finite-size effects for the most common ensembles of isotropic random matrices.
dc.affiliationpl
Wydział Fizyki, Astronomii i Informatyki Stosowanej : Instytut Fizyki im. Mariana Smoluchowskiego
dc.contributor.authorpl
Burda, Zdzisław - 127492
dc.contributor.authorpl
Livan, Giacomo
dc.contributor.authorpl
Święch, Artur
dc.date.accessioned
2015-06-30T14:18:06Z
dc.date.available
2015-06-30T14:18:06Z
dc.date.issuedpl
2013
dc.description.adminpl
[AB] Święch, Artur
dc.description.numberpl
2
dc.description.publicationpl
1
dc.description.volumepl
88
dc.identifier.articleidpl
022107
dc.identifier.doipl
10.1103/PhysRevE.88.022107
dc.identifier.eissnpl
1550-2376
dc.identifier.issnpl
1539-3755
dc.identifier.uri
http://ruj.uj.edu.pl/xmlui/handle/item/10807
dc.languagepl
eng
dc.language.containerpl
eng
dc.rights*
Dodaję tylko opis bibliograficzny
dc.rights.licence
bez licencji
dc.rights.uri*
dc.subtypepl
Article
dc.titlepl
Commutative law for products of infinitely large isotropic random matrices
dc.title.journalpl
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
dc.typepl
JournalArticle
dspace.entity.type
Publication
Affiliations

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