Critical behavior in Angelesco ensembles

TL;DR

This paper studies phase transitions in Angelesco ensembles with modified Jacobi weights, revealing a new universal kernel in a double scaling limit through Riemann-Hilbert analysis.

Contribution

It introduces the Angelesco kernel as a universal limit in a critical regime of Angelesco ensembles with touching intervals.

Findings

Identification of a new universal kernel, the Angelesco kernel K^{Ang}.

Demonstration of phase transition behavior as parameter a approaches -1.

Application of steepest descent analysis to multiple orthogonal polynomials.

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 \to -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 order n^{-1/2}. The correlation kernel converges, in this regime, to a new kind of universal kernel, the Angelesco kernel K^{Ang}. The result follows from the Deift/Zhou steepest descent analysis, applied to the Riemann-Hilbert problem for multiple orthogonal polynomials.