| 研究実績の概要 |
Recall that three classical beta ensembles on the real line are: Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles. For Gaussian beta ensembles, both static and dynamical results have been known. We focus on the other two ensembles.
In the so-called high temperature regime, the empirical measures of beta Laguerre ensembles are known to converge to a limiting measure which is the probability measure of associated Laguerre polynomials (Model II). We establish a dynamical version of that result. Namely, consider beta Laguerre processes in a high temperature regime, we show that their empirical measure processes converge to a limiting process (in probability). For the proof, we develop a moment method at the process level. The key ideas are: (i) each moment process of the empirical measure processes can be shown to converge to a deterministic process by induction, and (ii) under some additional mild conditions, the limiting moment processes determine the limiting measure uniquely, and thus, the convergence of the empirical measure processes follow.
We also extend the approach to study beta Jacobi ensembles and beta Jacobi processes at a high temperature regime. For the result, we obtain a new model of associated Jacobi polynomials, named Model III.
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