We study the fluctuations in the storage capacity of the symmetric binary perceptron, or equivalently, the fluctuations in the combinatorial discrepancy of a Gaussian matrix. Perkins and Xu, and Abbe, Li, and Sly recently established a sharp threshold: for some explicit constant $K_c$, the discrepancy of a Gaussian matrix is $K_c + o(1)$ with probability tending to one, but without a quantitative rate. We sharpen these results significantly. We show the fluctuations around $K_c$ of the discrepancy are at most of order $\log(n)/n$, and provide exponential tail bounds. Up to a logarithmic factor, this yields a tight characterization for the fluctuations of the symmetric perceptron and combinatorial discrepancy.

翻译：我们研究了对称二进制立方体的储存容量的波动,或相等于高斯矩阵组合差异的波动。 Perkins和Xu以及Abbe、Li和Sly最近设定了一个尖锐的阈值:对于某些明确的常数K美元,高斯矩阵的差异是K_c + o(1)美元,概率为1美元,但没有量化率。我们显著地放大了这些结果。我们显示了差异在K_c美元左右的波动,最多为$\log(n)/n美元,并且提供了指数尾线。根据对数系数,这产生了对正数的倍数和组合差异波动的严格定性。