Anticoncentration of the Permanent in Ginibre Ensembles

Abstract

Let $\mathbb{K}\in \{\mathbb{R},\mathbb{C},\mathbb{H}\}$, put $\beta ={\dim }_{\mathbb{R}}\mathbb{K}$, 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 $W_n^{\mathbb{K}}={\mathrm{per}}_{\mathbb{K}}{G}_n^{\mathbb{K}}/\sqrt{n!}$ has a radial density $p_n^{\mathbb{K}}$ satisfying $\Vert p_n^{\mathbb{K}}{\Vert }_{\infty }=p_n^{\mathbb{K}}(0){\lesssim }_{\beta }{n}^{(\beta +2)/4}$ and ${\sup }_{z\in \mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\le \epsilon ){\lesssim }_{\beta }{n}^{(\beta +2)/4}{\epsilon }^{\beta }$. 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.