On the Maximum of the Potential of a General Two-Dimensional Coulomb Gas
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929261529530368 |
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| author | Peilen, Luke |
| author_facet | Peilen, Luke |
| contents | We determine the leading order of the maximum of the random potential associated to a two-dimensional Coulomb gas for general $β$ and general confinement potential, extending the recent result of Lambert-Leblé-Zeitouni. In the case $β=2$, this corresponds to the (centered) log-characteristic polynomial of either the Ginibre random matrix ensemble for $V(x)=\frac{|x|^2}{2}$ or a more general normal matrix ensemble. The result on the leading order asymptotics for the maximum of the log-characteristic polynomial is new for random normal matrices. We rely on connections with the classical obstacle problem and the theory of Gaussian Multiplicative Chaos. We make use of a new concentration result for fluctuations of $C^{1,1}$ linear statistics which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00670 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Maximum of the Potential of a General Two-Dimensional Coulomb Gas Peilen, Luke Probability Mathematical Physics We determine the leading order of the maximum of the random potential associated to a two-dimensional Coulomb gas for general $β$ and general confinement potential, extending the recent result of Lambert-Leblé-Zeitouni. In the case $β=2$, this corresponds to the (centered) log-characteristic polynomial of either the Ginibre random matrix ensemble for $V(x)=\frac{|x|^2}{2}$ or a more general normal matrix ensemble. The result on the leading order asymptotics for the maximum of the log-characteristic polynomial is new for random normal matrices. We rely on connections with the classical obstacle problem and the theory of Gaussian Multiplicative Chaos. We make use of a new concentration result for fluctuations of $C^{1,1}$ linear statistics which may be of independent interest. |
| title | On the Maximum of the Potential of a General Two-Dimensional Coulomb Gas |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2403.00670 |