Mesoscopic transition for $β$-ensembles at intermediary temperature

This paper establishes a mesoscopic central limit theorem for linear statistics of $β$-ensembles or log-gas, as the dimension $N\to\infty$, in the temperature regime $1/N\llβ(N)\le 1$. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluct