The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal Matrices
The Dyson index, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, plays an essential role in random matrix theory, as it labels the so-calle...
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MDPI AG
2023-05-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/6/868 |
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| author | Cleverson A. Goulart Mauricio P. Pato |
| author_facet | Cleverson A. Goulart Mauricio P. Pato |
| author_sort | Cleverson A. Goulart |
| collection | DOAJ |
| description | The Dyson index, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, plays an essential role in random matrix theory, as it labels the so-called “three-fold way” that refers to the symmetries satisfied by ensembles under unitary transformations. As is known, its <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> 2, and 4 values denote the orthogonal, unitary, and symplectic classes, whose matrix elements are real, complex, and quaternion numbers, respectively. It functions, therefore, as a measure of the number of independent non-diagonal variables. On the other hand, in the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> ensembles, which represent the tridiagonal form of the theory, it can assume any real positive value, thus losing that function. Our purpose, however, is to show that, when the Hermitian condition of the real matrices generated with a given value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>β</mi><mo>.</mo></mrow></semantics></math></inline-formula> Therefore, it is as if the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> index were, in this way, again operative. It is shown that this effect happens for the three tridiagonal ensembles, namely, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>–Hermite, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>–Laguerre, and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>–Jacobi ensembles. |
| first_indexed | 2024-03-11T02:29:42Z |
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| language | English |
| last_indexed | 2024-03-11T02:29:42Z |
| publishDate | 2023-05-01 |
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| series | Entropy |
| spelling | doaj.art-6abc348f0d344ea9be157a45fa53d5802023-11-18T10:17:34ZengMDPI AGEntropy1099-43002023-05-0125686810.3390/e25060868The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal MatricesCleverson A. Goulart0Mauricio P. Pato1Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, São Paulo 05314-970, SP, BrazilInstituto de Física, Universidade de São Paulo, Caixa Postal 66318, São Paulo 05314-970, SP, BrazilThe Dyson index, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, plays an essential role in random matrix theory, as it labels the so-called “three-fold way” that refers to the symmetries satisfied by ensembles under unitary transformations. As is known, its <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> 2, and 4 values denote the orthogonal, unitary, and symplectic classes, whose matrix elements are real, complex, and quaternion numbers, respectively. It functions, therefore, as a measure of the number of independent non-diagonal variables. On the other hand, in the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> ensembles, which represent the tridiagonal form of the theory, it can assume any real positive value, thus losing that function. Our purpose, however, is to show that, when the Hermitian condition of the real matrices generated with a given value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>β</mi><mo>.</mo></mrow></semantics></math></inline-formula> Therefore, it is as if the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> index were, in this way, again operative. It is shown that this effect happens for the three tridiagonal ensembles, namely, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>–Hermite, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>–Laguerre, and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>–Jacobi ensembles.https://www.mdpi.com/1099-4300/25/6/868random matrix theory<i>β</i>-ensemblespseudo-HermitianPT-symmetry |
| spellingShingle | Cleverson A. Goulart Mauricio P. Pato The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal Matrices Entropy random matrix theory <i>β</i>-ensembles pseudo-Hermitian PT-symmetry |
| title | The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal Matrices |
| title_full | The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal Matrices |
| title_fullStr | The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal Matrices |
| title_full_unstemmed | The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal Matrices |
| title_short | The Double Dyson Index <i>β</i> Effect in Non-Hermitian Tridiagonal Matrices |
| title_sort | double dyson index i β i effect in non hermitian tridiagonal matrices |
| topic | random matrix theory <i>β</i>-ensembles pseudo-Hermitian PT-symmetry |
| url | https://www.mdpi.com/1099-4300/25/6/868 |
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