Phase transitions for products of characteristic polynomials under Dyson Brownian motion

We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $beta$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.