The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes

In a high temperature regime where $βN \to 2c$, the empirical distribution of the eigenvalues of Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles converges to a limiting measure which is related to associated Hermite polynomials, associated Laguerre polynomials and associated Jacobi polynomials, respectively. Here $β$ is the inverse temperature parameter, $N$ is the system size and $c>0$ is a given constant. This paper studies the spectral measure of the random tridiagonal matrix model of the three classical beta ensembles. We show that in the high temperature regime, the spectral measure converges in distribution to a Dirichlet process with base distribution being the limiting distribution, and scaling parameter $c$. Consequently, the spectral measure of a related semi-infinite Jacobi matrix coincides with that Dirichlet process, which provides examples of random Jacobi matrices with explicit spectral measures.