Limiting distributions of the largest eigenvalues of sparse matrices

Abstract

Random matrix theory, first considered by Eugene Wigner, has been applied in various areas of mathematics, physics and engineering. There are two important properties of the random matrix theory, called the universality. The first one is the bulk universality that concerns the behavior of eigenvalues in the bulk of the spectrum. The second one is about the limiting distribution of the largest eigenvalue that is called the edge universality. However, edge universality does not work for all the random matrices, such as the sparse matrices. This paper discusses the limiting distribution of the largest eigenvalue of the sparse matrices. According to early studies, under some suitable conditions, the limiting distributions of the largest eigenvalues of the sparse matrices have a deterministic shift from the Tracy-Widom distribution. We show that there is a correspondence between the theory and practice by making 1500 sample random matrices generated by \MATLAB. In addition, we suggest some reasons why the rescaled largest eigenvalue will not follow the Tracy-Widom distribution, or even the Gaussian distribution when $p \leq N^{-2/3}$.