Title: Extremal spectral behavior of weighted random $d$-regular graphs Show Chinese title

Title: 加权随机$d$-正则图的极值谱行为

Abstract: Analyzing the spectral behavior of random matrices with dependency among entries is a challenging problem. The adjacency matrix of the random $d$-regular graph is a prominent example that has attracted immense interest. A crucial spectral observable is the extremal eigenvalue, which reveals useful geometric properties of the graph. According to the Alon's conjecture, which was verified by Friedman, the (nontrivial) extremal eigenvalue of the random $d$-regular graph is approximately $2\sqrt{d-1}$. In the present paper, we analyze the extremal spectrum of the random $d$-regular graph (with $d\ge 3$ fixed) equipped with random edge-weights, and precisely describe its phase transition behavior with respect to the tail of edge-weights. In addition, we establish that the extremal eigenvector is always localized, showing a sharp contrast to the unweighted case where all eigenvectors are delocalized. Our method is robust and inspired by a sparsification technique developed in the context of Erd\H{o}s-R\'{e}nyi graphs (Ganguly and Nam, '22), which can also be applied to analyze the spectrum of general random matrices whose entries are dependent. Show Chinese abstract

Abstract: 分析具有条目之间依赖性的随机矩阵的谱行为是一个具有挑战性的问题。随机$d$正则图的邻接矩阵是一个突出的例子，引起了极大的兴趣。一个关键的谱可观测量是极值特征值，它揭示了图的有用几何特性。根据阿隆猜想，该猜想已被弗里德曼验证，随机$d$正则图的（非平凡）极值特征值大约为$2\sqrt{d-1}$。在本文中，我们分析了配备随机边权重的随机$d$正则图（$d\ge 3$固定）的极值谱，并精确描述了其相变行为，相对于边权重的尾部而言。此外，我们建立了极值特征向量总是局域化的，这与无权情况下所有特征向量都是非局域的形成鲜明对比。我们的方法具有鲁棒性，并受到在 Erdős-Rényi 图背景下开发的稀疏化技术的启发（Ganguly 和 Nam，'22），该技术也可以用于分析其条目相关的一般随机矩阵的谱。