| 研究実績の概要 |
We study dynamical versions of the three classical beta ensembles (Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles), that is, stochastic processes called beta Dyson’s Brownian motions, beta Laguerre processes and beta Jacobi processes. We develop a moment method at the process level to solve two fundamental problems: the convergence of the empirical measure process to a limiting process, and fluctuations around the limit. This approach works well for Gaussian and Laguerre cases, yielding dynamical versions of existing results on Gaussian beta ensembles and beta Laguerre ensembles. In addition, it provides a natural way to explain the appearance of orthogonal polynomials in the problem of Gaussian fluctuations around the limit.
Let us introduce results in the Gaussian case. In a high temperature regime, the empirical distribution of the eigenvalues converges weakly to the probability measures of associated Hermite polynomials. Gaussian fluctuations around the limit have been studied via the joint density and the tridiagonal random matrix model. Our new moment approach at the process level yields an analogous result to that in the regime where beta is fixed: by taking primitives of associated Hermite polynomials, Gaussian limits are independent.
It is worth noting that the approach works well in Gaussian and Laguerre cases in any regime where the parameter beta is allowed to vary as the system size tends to infinity. A little more work is needed to deal with the Jacobi case.
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| 今後の研究の推進方策 |
We are going to refine the moment method so that it can work for beta Jacobi processes as well. For that, we need some tools from the theory of gradient flows In addition, we plan to extend the method to study the freezing regime, the duality of the high temperature regime.
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