Anticoncentration of the Permanent in Ginibre Ensembles

2026-07-22

AI Summary

This study addresses the anticoncentration phenomenon of the permanent of complex Gaussian random matrices—specifically, the complex Ginibre ensemble—with the aim of verifying the Permanent Anticoncentration Conjecture proposed by Aaronson and Arkhipov. By establishing a comparison between the permanent and the squared Study determinant under the Laplace transform order, and combining radial density estimates with probabilistic inequalities, the authors provide the first proof that the normalized permanent in the complex setting satisfies an anticoncentration property. Their main contributions include deriving an upper bound of $O(n^{(\beta+2)/4})$ on the probability density and establishing a corresponding-order anticoncentration upper bound for the probability measure in any neighborhood of a point, thereby fully resolving the conjecture in the complex domain.

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.

Problem

• permanent
• anticoncentration
• Ginibre ensemble
• random matrix
• Gaussian

Innovation

• permanent anticoncentration
• Ginibre ensemble
• Laplace-transform order
• Study determinant
• random matrix theory