Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles
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2024
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| author | Forrester, Peter J. Kumar, Santosh |
| author_facet | Forrester, Peter J. Kumar, Santosh |
| contents | We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles. In the case $β=1$ this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for $β$ a positive integer, and the parameter $λ_1$ a non-negative integer. This case is special due to a finite basis of elementary functions, with coefficients which are polynomials. In the Laguerre case with $β= 1$ and $λ_1 + 1/2$ a non-negative integer some evidence is given of their again being a finite basis, now consisting of elementary functions and the error function multiplied by elementary functions. Moreover, from this the corresponding distributions in the fixed trace case permit a finite basis of power functions, as also for $λ_1$ a non-negative integer. The fixed trace case in this setting is relevant to quantum information theory and quantum transport problem, allowing particularly the exact determination of Landauer conductance distributions in a previously intractable parameter regime. Our findings also aid in analyzing zeros of the generating function for specific gap probabilities, supporting the validity of an associated large $N$ local central limit theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16069 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles Forrester, Peter J. Kumar, Santosh Mathematical Physics Mesoscale and Nanoscale Physics Probability Data Analysis, Statistics and Probability Computation 15B52, 60B20, 15A18, 33C45, 11B37, 62B10, 81P40 We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles. In the case $β=1$ this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for $β$ a positive integer, and the parameter $λ_1$ a non-negative integer. This case is special due to a finite basis of elementary functions, with coefficients which are polynomials. In the Laguerre case with $β= 1$ and $λ_1 + 1/2$ a non-negative integer some evidence is given of their again being a finite basis, now consisting of elementary functions and the error function multiplied by elementary functions. Moreover, from this the corresponding distributions in the fixed trace case permit a finite basis of power functions, as also for $λ_1$ a non-negative integer. The fixed trace case in this setting is relevant to quantum information theory and quantum transport problem, allowing particularly the exact determination of Landauer conductance distributions in a previously intractable parameter regime. Our findings also aid in analyzing zeros of the generating function for specific gap probabilities, supporting the validity of an associated large $N$ local central limit theorem. |
| title | Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles |
| topic | Mathematical Physics Mesoscale and Nanoscale Physics Probability Data Analysis, Statistics and Probability Computation 15B52, 60B20, 15A18, 33C45, 11B37, 62B10, 81P40 |
| url | https://arxiv.org/abs/2402.16069 |