A solution of the random eigenvalue problem by crossing theory

Abstract A method is developed for finding probabilistic descriptors of the eigenvalues of stochastic real-valued symmetric matrices. The method is based on the mean rate at which the characteristic polynomial of a stochastic matrix crosses level zero. The proposed approach can also be applied when additional information is available on the value of the first or the first few eigenvalues. Several examples are presented in the paper to demonstrate the application and usefulness of the method.