标题： 拉马努金性质和随机正则图的边普遍性 显示英文标题

标题： Ramanujan Property and Edge Universality of Random Regular Graphs

摘要： 我们考虑一个在$N$个顶点上的随机$d$-正则图的归一化邻接矩阵，其固定度数为$d\geq 3$，并将其特征值记为$\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$。我们在$N\rightarrow \infty$下建立以下两个结果。 (i) 高概率下，所有特征值除了额外的$N^{{\rm o}(1)}$因子外都是最优刚性的。 具体而言，块特征值的波动被$N^{-1+{\rm o}(1)}$限制，而边缘特征值的波动被$N^{-2/3+{\rm o}(1)}$限制。 (ii) 随机$d$-正则图满足边缘普适性。 也就是说，$\lambda_2$和$-\lambda_N$的分布收敛到与高斯正交系综相关的 Tracy-Widom$_1$分布。 因此，对于足够大的$N$，大约$69\%$的$d$-正则图在$N$个顶点上是Ramanujan，意味着$\max\{\lambda_2,|\lambda_N|\}\leq 2$。 显示英文摘要

摘要： We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We establish the following two results as $N\rightarrow \infty$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{{\rm o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+{\rm o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2/3+{\rm o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of $\lambda_2$ and $-\lambda_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max\{\lambda_2,|\lambda_N|\}\leq 2$.