Diffusion method in random matrix theory

We introduce a calculational tool useful in computing ratios and products of characteristic polynomials averaged over Gaussian measures with an external source. The method is based on Dyson's Brownian motion and Grassmann/ complex integration formulas for determinants. The resulting formulas are exact for finite matrix size N and form integral representations convenient for large N asymptotics. Quantities obtained by the method are interpreted as averages over standard matrix models. We provide several explicit and novel calculations with special emphasis on the beta = 2 Girko-Ginibre ensembles.