高斯格拉姆哈夫尼亚的精确矩揭示了弱反集中的$n^2/\log n$阈值
Exact Moments of Gaussian Gram Hafnians Reveal an $n^2/\log n$ Threshold for Weak Anticoncentration
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中文总结 AI 辅助
该研究通过计算高斯格拉姆哈夫尼亚的二阶、四阶矩，得到矩比的渐近行为，揭示了弱反集中的标度阶边界为$n^2/\log n$，为高斯玻色采样相关的难度论证提供了关键矩准则。
中文摘要 AI 辅助
反集中是近似采样难度论证的核心。在无碰撞高斯玻色采样的独立高斯替代模型中，本文研究的矩比还决定了平均理想线性交叉熵参考值。设$H_{k,n}=\mathrm{haf}(X^\mathsf{T}X)$，其中$X\in\mathbb{C}^{k\times 2n}$具有独立的标准圆复高斯元素。我们通过将四个哈夫尼亚副本约化为秩二高斯积分，精确计算了$\mathbb{E}|H_{k,n}|^2$和$\mathbb{E}|H_{k,n}|^4$。对于$R_{k,n}=(\mathbb{E}|H_{k,n}|^2)^2/\mathbb{E}|H_{k,n}|^4$，我们得到$R_{k,n}=4^{-n}\binom{2n}{n}/F_{k,n}$，其中$F_{k,n}={}_3F_2(-n,-n,1/2;1,k/2;1)$是一个终止的广义超几何多项式。若$k/n^2\to c>0$，则$F_{k,n}\to e^{1/c}I_0(1/c)$，其中$I_0$是零阶第一类修正贝塞尔函数，因此$R_{k,n}\sqrt{\pi n}\to[e^{1/c}I_0(1/c)]^{-1}$。由此，$k\asymp n^2$是一个平滑的贝塞尔交叉，而逆多项式弱反集中的标度阶边界为$k\asymp n^2/\log n$。这些结论涉及高斯替代矩准则；有限维哈尔矩转移和高概率小球反集中仍是独立问题。
英文摘要
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 $R_{k,n}=4^{-n}\binom{2n}{n}/F_{k,n}$, 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 $F{k,n}\to e^{1/c}I_0(1/c)$, where $I_0$ is the modified Bessel function of the first kind of order zero, and consequently $R_{k,n}\sqrt{πn}\to[e^{1/c}I_0(1/c)]^{-1}$. 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.
