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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.04661 |
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| _version_ | 1866917507339649024 |
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| author | Molag, L. D. |
| author_facet | Molag, L. D. |
| contents | We consider polynomial Bergman kernels with respect to exponentially varying weights $e^{-n \mathscr Q(z)}$ depending on a potential $\mathscr Q:\mathbb C^d\to\mathbb R$. We use these kernels to construct determinantal point processes on $\mathbb C^d$. Under mild conditions on the potential, the points are known to accumulate on a compact set $S_{\mathscr Q}$ called the droplet. We show that the local behavior of the kernel in the vicinity of the edge $\partial S_{\mathscr Q}$ is described in two different ways by universal limiting kernels. One of these limiting kernels is the error-function kernel, which is ubiquitous in random matrix theory, while the other limiting kernel is a new universal object: a multivariate version of the error-function kernel. We prove the universality in two qualitatively different settings: (i) the tensorized case where $\mathscr Q$ decomposes as a sum of planar potentials, and (ii) the case where $\mathscr Q$ is rotational symmetric. We also explicitly identify the subspace of the Bargmann-Fock space where the multivariate error-function kernel is reproducing. To treat regular edge points that exhibit a certain type of bulk degeneracy, we also find the behavior of the planar kernel with number of terms of order $o(n)$ instead of $n$. Lastly, we prove an edge scaling limit for counting statistics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04661 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A pluricomplex error-function kernel at the edge of polynomial Bergman kernels Molag, L. D. Probability Mathematical Physics Complex Variables 41A60, 60B20, 30E15, 15A52, 32U15, We consider polynomial Bergman kernels with respect to exponentially varying weights $e^{-n \mathscr Q(z)}$ depending on a potential $\mathscr Q:\mathbb C^d\to\mathbb R$. We use these kernels to construct determinantal point processes on $\mathbb C^d$. Under mild conditions on the potential, the points are known to accumulate on a compact set $S_{\mathscr Q}$ called the droplet. We show that the local behavior of the kernel in the vicinity of the edge $\partial S_{\mathscr Q}$ is described in two different ways by universal limiting kernels. One of these limiting kernels is the error-function kernel, which is ubiquitous in random matrix theory, while the other limiting kernel is a new universal object: a multivariate version of the error-function kernel. We prove the universality in two qualitatively different settings: (i) the tensorized case where $\mathscr Q$ decomposes as a sum of planar potentials, and (ii) the case where $\mathscr Q$ is rotational symmetric. We also explicitly identify the subspace of the Bargmann-Fock space where the multivariate error-function kernel is reproducing. To treat regular edge points that exhibit a certain type of bulk degeneracy, we also find the behavior of the planar kernel with number of terms of order $o(n)$ instead of $n$. Lastly, we prove an edge scaling limit for counting statistics. |
| title | A pluricomplex error-function kernel at the edge of polynomial Bergman kernels |
| topic | Probability Mathematical Physics Complex Variables 41A60, 60B20, 30E15, 15A52, 32U15, |
| url | https://arxiv.org/abs/2604.04661 |