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arXiv · 1203.2749

Critical behavior in Angelesco ensembles

K. Deschout·A. B. J. Kuijlaars
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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.

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K. Deschout, A. B. J. Kuijlaars. 2012-03-13. Critical behavior in Angelesco ensembles. https://doi.org/10.1063/1.4769822

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