| 研究実績の概要 |
Three classical beta ensembles on the real line, beta-Hermite, beta-Laguerre and beta-Jacobi ensembles, are realized as eigenvalues of three different types of random symmetric tridiagonal matrices. Note that symmetric tridiagonal matrices, also called Jacobi matrices, arise naturally in the theory of orthogonal polynomials on the real line. Indeed, from a Jacobi matrix, one forms a sequence of polynomials and under some conditions, there is a unique probability measure under which the polynomials are orthogonal. The unique probability measure is also referred to as the spectral measure of a Jacobi matrix.
When the parameter beta is fixed, the empirical distribution of eigenvalues of the three ensembles converges to the semi-circle distribution, Marchenko-Pastur distributions and Kesten-McKay distributions, respectively. These results can be intuitively seen via the random matrix models and spectral measures.
In this research, we found a further relation between classical beta ensembles and Hermite, Laguerre and Jacobi orthogonal polynomials. The relation reads as, in the regime where beta is proportional to the inverse of the system size, the empirical distribution converges to the probability measure of a corresponding associated Hermite, Laguerre and Jacobi orthogonal polynomials.
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