CLT for $\beta$-ensembles with Freud weights, application to the KLS conjecture in Schatten balls

In this paper, we are interested in the $\beta$-ensembles (or 1D log-gas) with Freud weights, namely with a potential of the form $|x|^{p}$ with $p \geq 2$. Since this potential is not of class $\mathcal{C}^{3}$ when $p \in (2,3]$, most of the literature does not apply. In this singular setting, we prove a central limit theorem for linear statistics with general test-functions. Our strategy relies on establishing an optimal local law in the spirit of [Bourgade, Mody, Pain 22'. Our results allow us to give a consistency check of the KLS conjecture for the uniform distributions on $p$-Schatten balls and the functions $f(X)=\mathrm{Tr}\left(X^r\right)^q$. While the case $p>3$, $q=1$, $r=2$ was proven in [Dadoun, Fradelizi, Gu\'edon, Zitt 23'], we address in the present paper the case $p\geq2$, $q\geq1$ and $r\geq2$ an even integer. The proofs are based on a link between the moments of norms of uniform laws on $p$-Schatten balls and the $\beta$-ensembles with Freud weights.