Characteristic Polynomials¶of Real Symmetric Random Matrices

The correlation functions of the random variables det(λ−X), in which X is an hermitian N×N random matrix, are known to exhibit universal local statistics in the large N limit. We study here the correlation of those same random variables for real symmetric matrices (GOE). The derivation relies on an exact dual representation of the problem: the k-point functions are expressed in terms of finite integrals over (quaternionic) k×k matrices. However the control of the Dyson limit, in which the distance of the various parameters λ's is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson–Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a finite number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large N limit.