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 $W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}$ 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.

Probability (math.PR); Computational Complexity (cs.CC); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

arXiv:2607.20329 [math.PR]

(or arXiv:2607.20329v1 [math.PR] for this version)

https://doi.org/10.48550/arXiv.2607.20329 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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