Localization of One-Dimensional Random Band Matrices

We consider a general class of n× n random band matrices with bandwidth W. When W² n, we prove that with high probability the eigenvectors of such matrices are localized and decay exponentially at the sharp scale W². Combined with the delocalization results of Yau and Yin [arXiv:2501.01718] and of Erdős and Riabov [arXiv:2506.06441], this establishes the conjectured localization-delocalization transition for a large class of random band matrices.