Improved discrepancy for the planar Coulomb gas at low temperatures
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916087646388224 |
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| author | Marceca, Felipe Romero, José Luis |
| author_facet | Marceca, Felipe Romero, José Luis |
| contents | We study the planar Coulomb gas in the regime where the inverse temperature $β_n$ grows at least logarithmically with respect to the number of particles $n$ (freezing regime, $β_n\gtrsim \log n$). We show that, almost surely for large $n$, the discrepancy between the number of particles in any microscopic region and their expected value (given with adequate precision by the equilibrium measure) is, up to log factors, of the order of the perimeter of the observation window. The estimates are valid throughout the whole droplet (the region where the particles accumulate), and are particularly interesting near the boundary, while in the bulk they offer technical improvements over known results.
Our work builds on recent results on equidistribution at low temperatures and improves on them by providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma dispersion kernel). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_14821 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Improved discrepancy for the planar Coulomb gas at low temperatures Marceca, Felipe Romero, José Luis Probability Mathematical Physics Functional Analysis 60K35, 82B26, 31C20, 47A75, 94A20 We study the planar Coulomb gas in the regime where the inverse temperature $β_n$ grows at least logarithmically with respect to the number of particles $n$ (freezing regime, $β_n\gtrsim \log n$). We show that, almost surely for large $n$, the discrepancy between the number of particles in any microscopic region and their expected value (given with adequate precision by the equilibrium measure) is, up to log factors, of the order of the perimeter of the observation window. The estimates are valid throughout the whole droplet (the region where the particles accumulate), and are particularly interesting near the boundary, while in the bulk they offer technical improvements over known results. Our work builds on recent results on equidistribution at low temperatures and improves on them by providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma dispersion kernel). |
| title | Improved discrepancy for the planar Coulomb gas at low temperatures |
| topic | Probability Mathematical Physics Functional Analysis 60K35, 82B26, 31C20, 47A75, 94A20 |
| url | https://arxiv.org/abs/2212.14821 |