Anticoncentration of the Permanent in Ginibre Ensembles

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 $W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}$ has a radial density $p_n^{\mathbb{K}}$ satisfying $\|p_n^{\mathbb{K}}\|_\infty=p_n^{\mathbb{K}}(0)\lesssim_\beta n^{(\beta+2)/4}$ and $\sup_{z\in\mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\leq\varepsilon)\lesssim_\beta n^{(\beta+2)/4}\varepsilon^\beta$.

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.