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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.12263 |
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| _version_ | 1866910714509131776 |
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| author | Krüger, Torben Lee, Seung-Yeop Yang, Meng |
| author_facet | Krüger, Torben Lee, Seung-Yeop Yang, Meng |
| contents | We study the normal matrix model, also known as the two-dimensional one-component plasma at a specific temperature, with merging singularity. As the number $n$ of particles tends to infinity we obtain the limiting local correlation kernel at the singularity, which is related to the parametrix of the Painlevé~II equation. The two main tools are Riemann-Hilbert problems and the generalized Christoffel-Darboux identity. The correlation kernel exhibits a novel anisotropic scaling behavior, where the corresponding spacing scale of particles is $n^{-1/3}$ in the direction of merging and $n^{-1/2}$ in the perpendicular direction. In the vicinity at different distances to the merging singularity we also observe Ginibre bulk and edge statistics, as well as the sine-kernel and the universality class corresponding to the elliptic ensemble in the weak non-Hermiticity regime for the local correlation function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12263 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local Statistics in Normal Matrix Models with Merging Singularity Krüger, Torben Lee, Seung-Yeop Yang, Meng Mathematical Physics We study the normal matrix model, also known as the two-dimensional one-component plasma at a specific temperature, with merging singularity. As the number $n$ of particles tends to infinity we obtain the limiting local correlation kernel at the singularity, which is related to the parametrix of the Painlevé~II equation. The two main tools are Riemann-Hilbert problems and the generalized Christoffel-Darboux identity. The correlation kernel exhibits a novel anisotropic scaling behavior, where the corresponding spacing scale of particles is $n^{-1/3}$ in the direction of merging and $n^{-1/2}$ in the perpendicular direction. In the vicinity at different distances to the merging singularity we also observe Ginibre bulk and edge statistics, as well as the sine-kernel and the universality class corresponding to the elliptic ensemble in the weak non-Hermiticity regime for the local correlation function. |
| title | Local Statistics in Normal Matrix Models with Merging Singularity |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2306.12263 |