For high-temperature 1D log gases, the rescaled largest particle obeys a large deviation principle with the iid rate function x^d - 1, and tridiagonal matrices with Gaussian tails obey the analogous principle with rate x^2 - 1.
Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform
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abstract
The aim of this paper is to identify the limit in a high temperature regime of classical beta ensembles on the real line and related eigenvalue processes by using the Markov--Krein transform. We show that the limiting measure of Gaussian beta ensembles (resp.\ beta Laguerre ensembles and beta Jacobi ensembles) is the inverse Markov--Krein transform of the Gaussian distribution (resp.\ the gamma distribution and the beta distribution). At the process level, we show that the limiting probability measure-valued process is the inverse Markov--Krein transform of a certain 1d stochastic process.
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Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature
For high-temperature 1D log gases, the rescaled largest particle obeys a large deviation principle with the iid rate function x^d - 1, and tridiagonal matrices with Gaussian tails obey the analogous principle with rate x^2 - 1.