Replica method for eigenvalues of real Wishart product matrices
We show how the replica method can be used to compute the asymptotic eigenvalue spectrum of a real Wishart product matrix. For unstructured factors, this provides a compact, elementary derivation of a polynomial condition on the Stieltjes transform first proved by Müller [IEEE Trans. Inf. Theory. 48...
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| Format: | Article |
| Language: | English |
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SciPost
2023-04-01
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| Series: | SciPost Physics Core |
| Online Access: | https://scipost.org/SciPostPhysCore.6.2.026 |
| _version_ | 1797851528689614848 |
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| author | Jacob A. Zavatone-Veth, Cengiz Pehlevan |
| author_facet | Jacob A. Zavatone-Veth, Cengiz Pehlevan |
| author_sort | Jacob A. Zavatone-Veth, Cengiz Pehlevan |
| collection | DOAJ |
| description | We show how the replica method can be used to compute the asymptotic eigenvalue spectrum of a real Wishart product matrix. For unstructured factors, this provides a compact, elementary derivation of a polynomial condition on the Stieltjes transform first proved by Müller [IEEE Trans. Inf. Theory. 48, 2086-2091 (2002)]. We then show how this computation can be extended to ensembles where the factors are drawn from matrix Gaussian distributions with general correlation structure. For both unstructured and structured ensembles, we derive polynomial conditions on the average values of the minimum and maximum eigenvalues, which in the unstructured case match the results obtained by Akemann, Ipsen, and Kieburg [Phys. Rev. E 88, 052118 (2013)] for the complex Wishart product ensemble. |
| first_indexed | 2024-04-09T19:19:20Z |
| format | Article |
| id | doaj.art-b3f37d7bf9a6477193d0a3f1aaa8ce3a |
| institution | Directory Open Access Journal |
| issn | 2666-9366 |
| language | English |
| last_indexed | 2024-04-09T19:19:20Z |
| publishDate | 2023-04-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics Core |
| spelling | doaj.art-b3f37d7bf9a6477193d0a3f1aaa8ce3a2023-04-05T16:16:54ZengSciPostSciPost Physics Core2666-93662023-04-016202610.21468/SciPostPhysCore.6.2.026Replica method for eigenvalues of real Wishart product matricesJacob A. Zavatone-Veth, Cengiz PehlevanWe show how the replica method can be used to compute the asymptotic eigenvalue spectrum of a real Wishart product matrix. For unstructured factors, this provides a compact, elementary derivation of a polynomial condition on the Stieltjes transform first proved by Müller [IEEE Trans. Inf. Theory. 48, 2086-2091 (2002)]. We then show how this computation can be extended to ensembles where the factors are drawn from matrix Gaussian distributions with general correlation structure. For both unstructured and structured ensembles, we derive polynomial conditions on the average values of the minimum and maximum eigenvalues, which in the unstructured case match the results obtained by Akemann, Ipsen, and Kieburg [Phys. Rev. E 88, 052118 (2013)] for the complex Wishart product ensemble.https://scipost.org/SciPostPhysCore.6.2.026 |
| spellingShingle | Jacob A. Zavatone-Veth, Cengiz Pehlevan Replica method for eigenvalues of real Wishart product matrices SciPost Physics Core |
| title | Replica method for eigenvalues of real Wishart product matrices |
| title_full | Replica method for eigenvalues of real Wishart product matrices |
| title_fullStr | Replica method for eigenvalues of real Wishart product matrices |
| title_full_unstemmed | Replica method for eigenvalues of real Wishart product matrices |
| title_short | Replica method for eigenvalues of real Wishart product matrices |
| title_sort | replica method for eigenvalues of real wishart product matrices |
| url | https://scipost.org/SciPostPhysCore.6.2.026 |
| work_keys_str_mv | AT jacobazavatonevethcengizpehlevan replicamethodforeigenvaluesofrealwishartproductmatrices |