Smallest gaps of the two-dimensional Coulomb gas
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914003661357056 |
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| author | Charlier, Christophe |
| author_facet | Charlier, Christophe |
| contents | We consider the two-dimensional Coulomb gas with a general potential at the determinantal temperature, or equivalently, the eigenvalues of random normal matrices. We prove that the smallest gaps between particles are typically of order $n^{-3/4}$, and that the associated joint point process of gap locations and gap sizes, after rescaling the gaps by $n^{3/4}$, converges to a Poisson point process. As a consequence, we show that the $k$-th smallest rescaled gap has a limiting density proportional to $x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^{4}}$, where $\mathcal{J}=π^{2}\int ρ(z)^{3}d^{2}z$ and $ρ$ is the density of the equilibrium measure. This generalizes a result of Shi and Jiang beyond the quadratic potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smallest gaps of the two-dimensional Coulomb gas Charlier, Christophe Probability Mathematical Physics We consider the two-dimensional Coulomb gas with a general potential at the determinantal temperature, or equivalently, the eigenvalues of random normal matrices. We prove that the smallest gaps between particles are typically of order $n^{-3/4}$, and that the associated joint point process of gap locations and gap sizes, after rescaling the gaps by $n^{3/4}$, converges to a Poisson point process. As a consequence, we show that the $k$-th smallest rescaled gap has a limiting density proportional to $x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^{4}}$, where $\mathcal{J}=π^{2}\int ρ(z)^{3}d^{2}z$ and $ρ$ is the density of the equilibrium measure. This generalizes a result of Shi and Jiang beyond the quadratic potential. |
| title | Smallest gaps of the two-dimensional Coulomb gas |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2507.23502 |