Author:

Keywords:

Science & Technology, Physical Sciences, Physics, Mathematical, Physics, GAUSSIAN RANDOM MATRICES, LARGE-N LIMIT, MULTIPLE ORTHOGONAL POLYNOMIALS, SQUARED BESSEL PATHS, EXTERNAL SOURCE, UNIVERSALITY, ASYMPTOTICS, math-ph, math.CA, math.MP, math.PR, 01 Mathematical Sciences, 02 Physical Sciences, Mathematical Physics, 49 Mathematical sciences, 51 Physical sciences

Abstract:

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 number 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. The result follows from the Deift/Zhou steepest descent analysis, applied to the Riemann-Hilbert problem for multiple orthogonal polynomials.