Fisher information among β-ensembles

In this paper, we consider the Fisher informations among three classical type β-ensembles when β > 0 scales with n satisfying \(\lim\limits_{n\to\infty}\beta n=\infty\) $$\underset{n\to \mathrm{\infty }}{lim}\beta n=\mathrm{\infty }$$ . We offer the exact order of the corresponding two Fisher informations, which indicates that the β-Laguerre ensembles do not satisfy the logarithmic Sobolev inequality. We also give some limit theorems on the extremals of β-Jacobi ensembles for β > 0 fixed.