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.

翻译：分析具有条目依赖关系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引起广泛关注的典型例子。一个关键的谱观测量是极值特征值，它揭示了图的有用几何性质。根据由Friedman验证的Alon猜想，随机$d$-正则图的（非平凡）极值特征值约为$2\sqrt{d-1}$。在本文中，我们分析了配备随机边权的随机$d$-正则图（其中$d\ge 3$固定）的极值谱，并精确描述了其相对于边权尾部的相变行为。此外，我们证明了极值特征向量总是局部化的，这与所有特征向量均为非局部化的无权重情况形成鲜明对比。我们的方法是稳健的，并受到在Erdős-Rényi图背景下发展的一种稀疏化技术的启发（Ganguly and Nam, '22），该技术也可用于分析条目具有依赖关系的一般随机矩阵的谱。