Random normal matrices: eigenvalue correlations near a hard wall
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916336458792960 |
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| author | Ameur, Yacin Charlier, Christophe Cronvall, Joakim |
| author_facet | Ameur, Yacin Charlier, Christophe Cronvall, Joakim |
| contents | We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant $Γ=2$ and that the number $n$ of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order $1/n$ from the hard edge. At distances much larger than $1/\sqrt{n}$, the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the ``semi-hard edge''.
More precisely, we provide asymptotics for the correlation kernel $K_{n}(z,w)$ as $n\to\infty$ in two microscopic regimes (with either $|z-w| = \mathcal{O} (1/\sqrt{n})$ or $|z-w| = \mathcal{O} (1/n)$), as well as in three macroscopic regimes (with $|z-w| \asymp 1$). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14166 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Random normal matrices: eigenvalue correlations near a hard wall Ameur, Yacin Charlier, Christophe Cronvall, Joakim Mathematical Physics Probability We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant $Γ=2$ and that the number $n$ of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order $1/n$ from the hard edge. At distances much larger than $1/\sqrt{n}$, the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the ``semi-hard edge''. More precisely, we provide asymptotics for the correlation kernel $K_{n}(z,w)$ as $n\to\infty$ in two microscopic regimes (with either $|z-w| = \mathcal{O} (1/\sqrt{n})$ or $|z-w| = \mathcal{O} (1/n)$), as well as in three macroscopic regimes (with $|z-w| \asymp 1$). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels. |
| title | Random normal matrices: eigenvalue correlations near a hard wall |
| topic | Mathematical Physics Probability |
| url | https://arxiv.org/abs/2306.14166 |