CLT for $β$-ensembles with Freud weights, application to the KLS conjecture in Schatten balls
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| Formato: | Preprint |
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2025
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| author | Guera, Charlie Dworaczek Memin, Ronan Pain, Michel |
| author_facet | Guera, Charlie Dworaczek Memin, Ronan Pain, Michel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05386 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | CLT for $β$-ensembles with Freud weights, application to the KLS conjecture in Schatten balls Guera, Charlie Dworaczek Memin, Ronan Pain, Michel Probability Functional Analysis Metric Geometry 60B20, 60F05, 52A23, 46B09 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. |
| title | CLT for $β$-ensembles with Freud weights, application to the KLS conjecture in Schatten balls |
| topic | Probability Functional Analysis Metric Geometry 60B20, 60F05, 52A23, 46B09 |
| url | https://arxiv.org/abs/2511.05386 |