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

In this paper, we are interested in the $β$-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, Guédon, 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 $β$-ensembles with Freud weights.

💡 Research Summary

The paper studies one‑dimensional log‑gases (β‑ensembles) with Freud weights V(x)=cₚ|x|ᵖ for p≥2, focusing on the regime where the potential is only C² and fails to be C³ at the origin (particularly p∈(2,3]). Classical results on local laws and central limit theorems (CLTs) for linear statistics require higher regularity, so the authors develop new techniques that work under this singular setting.

First, they define the β‑ensemble probability measure \