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On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials
, 2014
"... We consider multiple orthogonal polynomials with respect to Nikishin systems generated by two measures (σ1, σ2) with unbounded supports (suppσ1 ⊆ R+, suppσ2 ⊆ R−) and σ2 is discrete. A Nikishin type equilibrium problem in the presence of an external field acting on R+ and a constraint on R − is st ..."
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We consider multiple orthogonal polynomials with respect to Nikishin systems generated by two measures (σ1, σ2) with unbounded supports (suppσ1 ⊆ R+, suppσ2 ⊆ R−) and σ2 is discrete. A Nikishin type equilibrium problem in the presence of an external field acting on R+ and a constraint on R − is stated and solved. The solution is used for deriving the contracted zero distribution of the associated multiple orthogonal polynomials.
Critical behavior in Angelesco ensembles
, 2014
"... 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 ensemb ..."
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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 KAng. The result follows from the Deift/Zhou steepest descent analysis, applied to the RiemannHilbert 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.