| 研究実績の概要 |
Gaussian beta ensembles, the natural generalization of Gaussian orthogonal/unitary/symplectic ensembles, can be realized as eigenvalues of certain random Jacobi matrices with independent entries. The parameter beta here is regarded as the inverse temperature. Gaussian beta ensembles, for fixed beta, have been studied extensively. Many results in three main regimes: global, local/bulk, and edge regimes have been established. This research, however, aims to understand the dependence on beta of the ensembles. We get some results on the global regime, the regime which mainly concerns with the limiting behavior of the empirical distribution of the eigenvalues.
When beta is fixed, a well-known result in the global regime is Wigner's semi-circle law which states that the distribution of eigenvalues, chosen randomly, converges to the semi-circle distribution as the system size tends to infinity. In other words, this means that the empirical distribution of the eigenvalues converges weakly to the semi-circle distribution, almost surely.
When the inverse temperature beta is a function of the system size, the limit is no longer the semi-semicircle distribution. In this research, we can completely describe the limit of the empirical distribution in all cases. Besides the convergence to a limit, the fluctuation around the limit is also investigated by a new method which is applicable to a large class of Jacobi matrices with independent entries.
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| 今後の研究の推進方策 |
Three classical beta ensembles in the real line are Gaussian, Wishart and Jacobi beta ensembles. Although they are all realized as eigenvalues of random Jacobi matrices, the structures of the random matrices are all different. So far, several new results have been established for Gaussian beta ensembles and their related class of random Jacobi matrices. The next plan is to study analogous problems for Wishart and Jacobi beta ensembles and more general class of random Jacobi matrices.
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