Anticoncentration of the Permanent in Ginibre Ensembles

• URL: http://arxiv.org/abs/2607.20329v1
• Date: Wed, 22 Jul 2026 16:15:28 GMT
• Title: Anticoncentration of the Permanent in Ginibre Ensembles
• Abstract summary: We prove that the normalized row-ordered permanent $W_nmathbbK=nameper_mathbbKG_nmathbbK/sqrtn!$ has a radial density $p_nmathbbK$ satisfying $|p_nmathbbK|_infty=p_nmathbbK(+2)/4$ and $sup_zinmathbbKmathbbP(|W_
• Score: 11.09360259927697
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, put $β=\dim_{\mathbb{R}}\mathbb{K}$, and let $G_n^{\mathbb{K}}$ be an $n\times n$ matrix with i.i.d. standard $\mathbb{K}$-Gaussian entries, namely a standard $\mathbb{K}$-Ginibre matrix. We prove that the normalized row-ordered permanent $W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}$ has a radial density $p_n^{\mathbb{K}}$ satisfying $\|p_n^{\mathbb{K}}\|_\infty=p_n^{\mathbb{K}}(0)\lesssim_βn^{(β+2)/4}$ and $\sup_{z\in\mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\leq\varepsilon)\lesssim_βn^{(β+2)/4}\varepsilon^β$. In particular, for $\mathbb{K}=\mathbb{C}$, this resolves the Permanent Anticoncentration Conjecture of Aaronson and Arkhipov. The proof compares the squared Gaussian permanent with the squared (Study) determinant in Laplace-transform order.

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