Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been previously applied to sparse matrix problems. We close this gap in the literature by showing how one can employ numerical solutions of Hammerstein equations to accurately recover the spectra of adjacency matrices and Laplacians of random graphs. While our treatment focuses on random graphs for concreteness, the methodology has broad applications to more general sparse random matrices.

翻译：对于若干稀疏随机矩阵系综，其特征值分布的求解可归结为 Hammerstein 型非线性积分方程的求解。尽管此类方程已有系统的数学理论，但此前尚未应用于稀疏矩阵问题。我们通过展示如何利用 Hammerstein 方程的数值解来精确恢复随机图的邻接矩阵与拉普拉斯矩阵的谱，填补了文献中的这一空白。虽然本文为具体起见以随机图为主要研究对象，但该方法可广泛推广至更一般的稀疏随机矩阵。