标题： 非均匀随机矩阵的边缘谱形普适性 显示英文标题

标题： Edge Universality for Inhomogeneous Random Matrices

摘要： 我们考虑其元素为独立且对称的随机变量，且方差模式任意的对称和厄米特随机矩阵。 在一种新的短到长混合条件之下，该条件在意义上是尖锐的，因为它排除了光谱边缘处的修正位移，我们建立了此类非均匀随机矩阵的GOE/GUE边缘普适性。 此条件有效地将普适性问题转化为验证由方差轮廓矩阵所支配的随机游走的混合性质。 我们的普适性结果适用于一个显著广泛的随机矩阵集合，这些集合可能高度不均匀、稀疏或远超出经典随机矩阵理论的平均场设定。 显著的例子包括： 1. 非均匀Wishart型随机矩阵； 2. 其元素为独立随机变量且具有通用方差轮廓的随机带状矩阵，特别是在维度$d \le 2$中具有最优带宽； 3. 具有结构化方差轮廓的稀疏随机矩阵； 4. 在显著更弱的稀疏性约束和重尾元素分布下的一般Wigner矩阵； 5. 在尖锐混合假设下的Wegner轨道模型； 6. 随机$d$-正则图的随机2-提升，其中$d\geq N^{2/3+\epsilon}$对于任何$\epsilon>0$。 显示英文摘要

摘要： We consider symmetric and Hermitian random matrices whose entries are independent and symmetric random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition, which is sharp in the sense that it precludes a corrected shift at the spectral edge, we establish GOE/GUE edge universality for such inhomogeneous random matrices. This condition effectively reduces the universality problem to verifying the mixing properties of a random walk governed by the variance profile matrix. Our universality results are applicable to a remarkably broad class of random matrix ensembles that may be highly inhomogeneous, sparse, or far beyond the mean-field setting of classical random matrix theory. Notable examples include: 1. Inhomogeneous Wishart-type random matrices; 2. Random band matrices whose entries are independent random variables with general variance profile, particularly with an optimal bandwidth in dimensions $d \le 2$; 3. Sparse random matrices with structured variance profiles; 4. Generalized Wigner matrices under significantly weaker sparsity constraints and heavy-tailed entry distributions; 5. Wegner orbital models under sharp mixing assumptions; 6. Random 2-lifts of random $d$-regular graphs where $d\geq N^{2/3+\epsilon}$ for any $\epsilon>0$.