Commutative law for products of infinitely large isotropic random matrices

2013
journal article
article
8
 cris.lastimport.wos 2024-04-10T00:09:07Z dc.abstract.en 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. pl dc.affiliation Wydział Fizyki, Astronomii i Informatyki Stosowanej : Instytut Fizyki im. Mariana Smoluchowskiego pl dc.contributor.author Burda, Zdzisław - 127492 pl dc.contributor.author Livan, Giacomo pl dc.contributor.author Święch, Artur pl dc.date.accessioned 2015-06-30T14:18:06Z dc.date.available 2015-06-30T14:18:06Z dc.date.issued 2013 pl dc.description.admin [AB] Święch, Artur pl dc.description.number 2 pl dc.description.publication 1 pl dc.description.volume 88 pl dc.identifier.articleid 022107 pl dc.identifier.doi 10.1103/PhysRevE.88.022107 pl dc.identifier.eissn 1550-2376 pl dc.identifier.issn 1539-3755 pl dc.identifier.uri http://ruj.uj.edu.pl/xmlui/handle/item/10807 dc.language eng pl dc.language.container eng pl dc.rights Dodaję tylko opis bibliograficzny * dc.rights.licence bez licencji dc.rights.uri * dc.subtype Article pl dc.title Commutative law for products of infinitely large isotropic random matrices pl dc.title.journal Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics pl dc.type JournalArticle pl dspace.entity.type Publication
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
[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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