A circuit $\mathcal{C}$ samples a distribution $\mathbf{X}$ with an error $\epsilon$ if the statistical distance between the output of $\mathcal{C}$ on the uniform input and $\mathbf{X}$ is $\epsilon$. We study the hardness of sampling a uniform distribution over the set of $n$-bit strings of Hamming weight $k$ denoted by $\mathbf{U}^n_k$ for _decision forests_, i.e. every output bit is computed as a decision tree of the inputs. For every $k$ there is an $O(\log n)$-depth decision forest sampling $\mathbf{U}^n_k$ with an inverse-polynomial error [Viola 2012, Czumaj 2015]. We show that for every $\epsilon > 0$ there exists $\tau$ such that for decision depth $\tau \log (n/k) / \log \log (n/k)$, the error for sampling $\mathbf{U}_k^n$ is at least $1-\epsilon$. Our result is based on the recent robust sunflower lemma [Alweiss, Lovett, Wu, Zhang 2021, Rao 2019]. Our second result is about matching a set of $n$-bit strings with the image of a $d$-_local_ circuit, i.e. such that each output bit depends on at most $d$ input bits. We study the set of all $n$-bit strings whose Hamming weight is at least $n/2$. We improve the previously known locality lower bound from $\Omega(\log^* n)$ [Beyersdorff, Datta, Krebs, Mahajan, Scharfenberger-Fabian, Sreenivasaiah, Thomas and Vollmer, 2013] to $\Omega(\sqrt{\log n})$, leaving only a quartic gap from the best upper bound of $O(\log^2 n)$.

翻译：电路 $\mathcal{C}$ 以误差 $\epsilon$ 采样分布 $\mathbf{X}$，若其在均匀输入上的输出与 $\mathbf{X}$ 之间的统计距离为 $\epsilon$。我们研究在决策森林（即每个输出比特被计算为输入决策树）条件下，对由汉明重量为 $k$ 的 $n$ 比特字符串所构成的集合上的均匀分布 $\mathbf{U}^n_k$ 进行采样的难度。对于每个 $k$，存在深度为 $O(\log n)$ 的决策森林以逆多项式误差采样 $\mathbf{U}^n_k$ [Viola 2012, Czumaj 2015]。我们证明：对于任意 $\epsilon > 0$，存在 $\tau$ 使得当决策深度为 $\tau \log (n/k) / \log \log (n/k)$ 时，采样 $\mathbf{U}_k^n$ 的误差至少为 $1-\epsilon$。该结果基于近期鲁棒向日葵引理 [Alweiss, Lovett, Wu, Zhang 2021, Rao 2019]。我们的第二个结果涉及将一组 $n$ 比特字符串与 $d$-局部电路（即每个输出比特至多依赖于 $d$ 个输入比特）的像匹配。我们研究所有汉明重量至少为 $n/2$ 的 $n$ 比特字符串集合，将此前已知的局部性下界从 $\Omega(\log^* n)$ [Beyersdorff, Datta, Krebs, Mahajan, Scharfenberger-Fabian, Sreenivasaiah, Thomas and Vollmer, 2013] 改进至 $\Omega(\sqrt{\log n})$，仅与最优上界 $O(\log^2 n)$ 存在四次方差距。