Anticoncentration of the Permanent in Ginibre Ensembles

2026/07/22 by Frederic Koehler, Pui Kuen Leung
#math.PR #cs.CC #math-ph #math.MP #quant-ph

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Abstract

Let \mathbbK∈\ℝ,ℂ,ℍ\, put β=dim_ℝ\mathbbK, and let G_n^\mathbbK be an n× n matrix with i.i.d. standard \mathbbK-Gaussian entries, namely a standard \mathbbK-Ginibre matrix. We prove that the normalized row-ordered permanent W_n^\mathbbK=per_\mathbbKG_n^\mathbbK/√(n!) has a radial density p_n^\mathbbK satisfying ‖p_n^\mathbbK‖_∞=p_n^\mathbbK(0)\lesssim_βn^(β+2)/4 and sup_z∈\mathbbKℙ(|W_n^\mathbbK-z|≤ε)\lesssim_βn^(β+2)/4ε^β. In particular, for \mathbbK=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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