| 研究実績の概要 |
This research studies spectral properties of beta ensembles and related random matrix models in case the inverse temperature beta is allowed to vary with the system size. We obtain the following results.
(1) For beta Laguerre ensembles, one of three classical beta ensembles on the real line, we completely describe the global asymptotic behavior of the empirical distribution, that is, the convergence to a limit distribution and Gaussian fluctuations around the limit. Beta Laguerre ensembles are generalizations of the distribution of the eigenvalues of Wishart matrices or Laguerre matrices, two types of random matrices in statistics, in terms of the joint density. They are now realized as eigenvalues of a random tridiagonal matrix model. For the proof, we make use of the random matrix model and extend some ideas used in the case of Gaussian beta ensembles, another classical beta ensembles.
(2) For general beta ensembles on the real line in a high temperature regime, the regime where beta tends to zero at the rate of the reciprocal of the system size, we show that the local statistics around any fixed reference energy converges to a homogeneous Poisson point process. We prove the Poisson statistics by analyzing the joint density with the help of some estimates from the theory of large deviation principle.
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