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Number theorists have studied extensively the connections between the distribution of zeros of the Riemann ζ-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power K 2 ; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.

We consider a Hamiltonian H which is the sum of a deterministic part H 0 and of a random potential V . For finite N × N matrices, following a method introduced by Kazakov, we derive a representation of the correlation functions in terms of contour integrals over a finite number of variables. This allows one to analyse the level correlations, whereas the standard methods of random matrix theory, such as the method of orthogonal polynomials, are not available for such cases. At short distance we recover, for an arbitrary H 0 , an oscillating behavior for the connected two-level correlation.

The HarishChandra-Itzykson-Zuber integral over the unitary group U(k) (β = 2) is present in numerous problems involving Hermitian random matrices. It is well known that the result is semi-classically exact. This simple result does not extend to other symmetry groups, such as the symplectic or orthogonal groups. In this article the analysis of this integral is extended first to the symplectic group Sp(k) (β=4). There the semi-classical approximation has to be corrected by a WKB expansion. It turns out that this expansion stops after a finite number of terms ; in other words the WKB approximation is corrected by a polynomial in the appropriate variables. The analysis is based upon new solutions to the heat kernel differential equation. We have also investigated arbitrary values of the parameter β, which characterizes the symmetry group. Closed formulae are derived for arbitrary β and k = 3, and also for large β and arbitrary k.

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