Characteristic polynomials of random matrices

Number theorists have studied extensively the connections between the distr ibution of zeros of the Riemann zeta -function, and of some generalizations , with the statistics of the eigenvalues of large random matrices. It is in teresting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values o f the characteristic polynomials of the matrix. It turns out that these exp ectation values are quite interesting. For instance, the moments of order 2 K scale, for unitary invariant ensembles, as the density of eigenvalues rai sed 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 the ory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators o f 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.