Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $\beta$ ensembles
Abstract
We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $\beta$ ensembles. In the case $\beta=1$ this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for $\beta$ a positive integer, and the parameter $\lambda_1$ a nonnegative integer. This case is special due to a finite basis of elementary functions, with coefficients which are polynomials. In the Laguerre case with $\beta = 1$ and $\lambda_1 + 1/2$ a nonnegative 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 $\lambda_1$ a nonnegative 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.16069
 arXiv:
 arXiv:2402.16069
 Bibcode:
 2024arXiv240216069F
 Keywords:

 Mathematical Physics;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 Mathematics  Probability;
 Physics  Data Analysis;
 Statistics and Probability;
 Statistics  Computation;
 15B52;
 60B20;
 15A18;
 33C45;
 11B37;
 62B10;
 81P40
 EPrint:
 25 pages, 6 figures