We study the critical window of the symmetric binary perceptron, or equivalently, combinatorial discrepancy. Consider the problem of finding a binary vector $\sigma$ satisfying $\|A\sigma\|_\infty \le K$, where $A$ is an $\alpha n \times n$ matrix with iid Gaussian entries. For fixed $K$, at which densities $\alpha$ is this constraint satisfaction problem (CSP) satisfiable? A sharp threshold was recently established by Perkins and Xu, and Abbe, Li, and Sly , answering this to first order. Namely, for each $K$ there exists an explicit critical density $\alpha_c$ so that for any fixed $\epsilon > 0$, with high probability the CSP is satisfiable for $\alpha n < (\alpha_c - \epsilon ) n$ and unsatisfiable for $\alpha n > (\alpha_c + \epsilon) n$. This corresponds to a bound of $o(n)$ on the size of the critical window. We sharpen these results significantly, as well as provide exponential tail bounds. Our main result is that, perhaps surprisingly, the critical window is actually at most $O(\log n)$. More precisely, with high probability the CSP is satisfiable for $\alpha n < \alpha_c n -O(\log n)$ and unsatisfiable for any $\alpha n > \alpha_c n + \omega(1)$. This implies the symmetric perceptron has nearly the "sharpest possible transition," adding it to a short list of CSP for which the critical window is rigorously known to be of near-constant width.

翻译：我们研究了对称二元感知器（等价于组合差异问题）的临界窗口。考虑寻找满足 $\|A\sigma\|_\infty \le K$ 的二元向量 $\sigma$ 的问题，其中 $A$ 是一个元素独立同分布于标准高斯分布的 $\alpha n \times n$ 矩阵。对于固定的 $K$，该约束满足问题在何种密度 $\alpha$ 下可满足？Perkins 与 Xu 以及 Abbe、Li 与 Sly 最近建立了一个尖锐阈值，给出了该问题的一阶回答：对于每个 $K$，存在一个显式临界密度 $\alpha_c$，使得对任意固定 $\epsilon > 0$，当 $\alpha n < (\alpha_c - \epsilon)n$ 时约束满足问题高概率可满足，而当 $\alpha n > (\alpha_c + \epsilon)n$ 时高概率不可满足。这对应临界窗口大小为 $o(n)$。我们显著改进了这些结果，并给出了指数型尾部界。我们的主要结果是：令人意外的是，临界窗口实际上至多为 $O(\log n)$。更精确地说，当 $\alpha n < \alpha_c n - O(\log n)$ 时约束满足问题高概率可满足，而当 $\alpha n > \alpha_c n + \omega(1)$ 时高概率不可满足。这表明对称感知器具有近乎"最尖锐的相变"，使其成为临界窗口被严格证明为近似恒定宽度的少数约束满足问题之一。