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 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 $\beta = 1$ and $\lambda_1 + 1/2$ 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 $\lambda_1$ 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.
- Publication:
-
arXiv e-prints
- 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
- E-Print:
- 25 pages, 6 figures