Spectral properties of the generalized diluted Wishart ensemble
The celebrated Marčenko–Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple fo...
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| Format: | Article |
| Language: | English |
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IOP Publishing
2022-01-01
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| Series: | Journal of Physics: Complexity |
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| Online Access: | https://doi.org/10.1088/2632-072X/ac956d |
| _version_ | 1811158059859312640 |
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| author | Isaac Pérez Castillo |
| author_facet | Isaac Pérez Castillo |
| author_sort | Isaac Pérez Castillo |
| collection | DOAJ |
| description | The celebrated Marčenko–Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as $\boldsymbol{F}\equiv \frac{1}{2d}\left(\boldsymbol{X}{\boldsymbol{Y}}^{T}+\boldsymbol{Y}{\boldsymbol{X}}^{T}\right)$ , where X and Y are diluted N × P rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution $P({x}_{i}^{\mu },{y}_{i}^{\mu })=\frac{d}{N}\varrho ({x}_{i}^{\mu },{y}_{i}^{\mu })+\left(1-\frac{d}{N}\right){\delta }_{{x}_{i}^{\mu },0}{\delta }_{{y}_{i}^{\mu },0}$ , with the probability density ϱ ( x , y ) controlling the correlation between the matrices entries of X and Y . Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph d , the rectangularity of the matrices α = N / P , and the degree of correlation of the matrix entries via the density ϱ ( x , y ). Finally, we compare our findings to numerical diagonalisation showing excellent agreement. |
| first_indexed | 2024-04-10T05:17:55Z |
| format | Article |
| id | doaj.art-348b4e3c722149c9aefb4b02c63c711b |
| institution | Directory Open Access Journal |
| issn | 2632-072X |
| language | English |
| last_indexed | 2024-04-10T05:17:55Z |
| publishDate | 2022-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | Journal of Physics: Complexity |
| spelling | doaj.art-348b4e3c722149c9aefb4b02c63c711b2023-03-08T15:04:41ZengIOP PublishingJournal of Physics: Complexity2632-072X2022-01-013404500110.1088/2632-072X/ac956dSpectral properties of the generalized diluted Wishart ensembleIsaac Pérez Castillo0https://orcid.org/0000-0001-7622-9440Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa , San Rafael Atlixco 186, Ciudad de México 09340, MexicoThe celebrated Marčenko–Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as $\boldsymbol{F}\equiv \frac{1}{2d}\left(\boldsymbol{X}{\boldsymbol{Y}}^{T}+\boldsymbol{Y}{\boldsymbol{X}}^{T}\right)$ , where X and Y are diluted N × P rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution $P({x}_{i}^{\mu },{y}_{i}^{\mu })=\frac{d}{N}\varrho ({x}_{i}^{\mu },{y}_{i}^{\mu })+\left(1-\frac{d}{N}\right){\delta }_{{x}_{i}^{\mu },0}{\delta }_{{y}_{i}^{\mu },0}$ , with the probability density ϱ ( x , y ) controlling the correlation between the matrices entries of X and Y . Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph d , the rectangularity of the matrices α = N / P , and the degree of correlation of the matrix entries via the density ϱ ( x , y ). Finally, we compare our findings to numerical diagonalisation showing excellent agreement.https://doi.org/10.1088/2632-072X/ac956dWishart ensemblespectral densitycross correlation matricescavity method |
| spellingShingle | Isaac Pérez Castillo Spectral properties of the generalized diluted Wishart ensemble Journal of Physics: Complexity Wishart ensemble spectral density cross correlation matrices cavity method |
| title | Spectral properties of the generalized diluted Wishart ensemble |
| title_full | Spectral properties of the generalized diluted Wishart ensemble |
| title_fullStr | Spectral properties of the generalized diluted Wishart ensemble |
| title_full_unstemmed | Spectral properties of the generalized diluted Wishart ensemble |
| title_short | Spectral properties of the generalized diluted Wishart ensemble |
| title_sort | spectral properties of the generalized diluted wishart ensemble |
| topic | Wishart ensemble spectral density cross correlation matrices cavity method |
| url | https://doi.org/10.1088/2632-072X/ac956d |
| work_keys_str_mv | AT isaacperezcastillo spectralpropertiesofthegeneralizeddilutedwishartensemble |