论文标题
定量的tracy-widom定律，用于广义Wigner矩阵的最大特征值
Quantitative Tracy-Widom laws for the largest eigenvalue of generalized Wigner matrices
论文作者
论文摘要
我们表明，随着矩阵dimension $ n $倾向于无限，任何广义的Wigner矩阵$ h $中最大的特征值的波动汇总到Tracy-Widom法律。我们允许$ h $的条目的差异具有不同的值，但大小具有可比的大小，因此$ \ sum_ {i} \ mathbb {e} | h_ {ij} |^|^2 = 1 $。我们的结果改善了Bourgade [8]的先前速率$ O（N^{ - 2/9}）$，证明依赖于第一个长期绿色功能比较定理在边缘附近，而没有第二次匹配限制。
We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly $O(N^{-1/3})$, as the matrix dimension $N$ tends to infinity. We allow the variances of the entries of $H$ to have distinct values but of comparable sizes such that $\sum_{i} \mathbb{E}|h_{ij}|^2=1$. Our result improves the previous rate $O(N^{-2/9})$ by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.
