  It is shown that the correlation functions of the random variables
$\det(\lambda - X)$, in which $X$ is a real symmetric $ N\times N$ random
matrix, exhibit universal local statistics in the large $N$ limit. The
derivation relies on an exact dual representation of the problem: the $k$-point
functions are expressed in terms of finite integrals over (quaternionic)
$k\times k$ matrices. However the control of the Dyson limit, in which the
distance of the various parameters $\la$'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 {\it 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.
