Characteristic polynomials of real symmetric random matrices

The correlation functions of the random variables det(lambda - X), in which X is an hermitian N x N random matrix, are known to exhibit universal loca l statistics in the large N limit. We study here the correlation of those s ame random variables for real symmetric matrices (GOE). The derivation reli es on an exact dual representation of the problem: the k-point functions ar e expressed in terms of finite integrals over (quaternionic) k x k matrices . However the control of the Dyson limit, in which the distance of the vari ous parameters X's is of the order of the mean spacing, requires an integra tion over the symplectic group. It is shown that a generalization of the It zykson-Zuber method holds for this problem, but contrary to the unitary cas e, the semi-classical result requires a finite number of corrections to be exact. We have also considered the problem of an external matrix source cou pled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large N limit.