Critical behavior in Angelesco ensembles

We consider Angelesco ensembles with respect to two modified Jacobi weights on touching intervals [a, 0] and [0, 1], for a < 0. As a → −1 the particles around 0 experience a phase transition. This transition is studied in a double scaling limit, where we let the num-ber of particles of the ensemble tend to infinity while the parameter a tends to −1 at a rate of O(n−1/2). The correlation kernel converges, in this regime, to a new kind of universal kernel, the Angelesco kernel KAng. The result follows from the Deift/Zhou steepest descent anal-ysis, applied to the Riemann-Hilbert problem for multiple orthogonal polynomials. 1 Introduction and statement of results Multiple orthogonal polynomial (MOP) ensembles [21] form an extension of the more familiar orthogonal polynomial (OP) ensembles [20]. The latter appear as the eigenvalue distributions of unitary random matrix ensembles