Exact Moments of Gaussian Gram Hafnians Reveal an $n^2/\log n$ Threshold for Weak Anticoncentration

Abstract

Anticoncentration is central to hardness arguments for approximate sampling. In the independent Gaussian surrogate for collision free Gaussian boson sampling, the moment ratio studied here also determines the averaged ideal linear cross entropy reference value. Let $H_{k,n}=\mathrm{haf}(X^{\mathsf{T}}X)$, where $X\in {\mathbb{C}}^{k\times 2n}$ has independent standard circular complex Gaussian entries. We evaluate $\mathbb{E}|H_{k,n}{|}^2$ and $\mathbb{E}|H_{k,n}{|}^4$ exactly by reducing four hafnian copies to a rank two Gaussian integral. For $R_{k,n}=(\mathbb{E}|H_{k,n}{|}^2{)}^2/\mathbb{E}|H_{k,n}{|}^4$, we obtain $R_{k,n}=4^{-n}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)/F_{k,n}$, where $F_{k,n}=3F_2(-n,-n,1/2;1,k/2;1)$ is a terminating generalized hypergeometric polynomial. If $k/n^2\to c>0$, then $Fk,n\to e^{1/c}{I}_0(1/c)$, where $I_0$ is the modified Bessel function of the first kind of order zero, and consequently $R_{k,n}\sqrt{\pi n}\to [e^{1/c}{I}_0(1/c){]}^{-1}$. Thus $k\asymp n^2$ is a smooth Bessel crossover, whereas the scaling order boundary for inverse polynomial weak anticoncentration is $k\asymp n^2/\log n$. These conclusions concern the Gaussian surrogate moment criterion; finite dimensional Haar moment transfer and high probability small ball anticoncentration remain separate problems.

15 pages, 1 figure. Includes the main paper, End Matter, and Supplemental Material as appendices. An accompanying Lean 4 verification package is available from Zenodo at https://doi.org/10.5281/zenodo.21959946