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Published March 2011 | public
Journal Article

A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

Abstract

We consider the two sequences of biorthogonal polynomials (p_(k,n))^∞_(k=0) and (q_(k,n)) ^∞_(k=0) related to the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p_(n,n) as n → ∞. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behaviour for a certain negative value of t. We also prove a general result about the interlacing of zeros of biorthogonal polynomials.

Additional Information

© 2011 IOP Publishing Ltd & London Mathematical Society. Received 6 July 2010. Published 11 February 2011. Recommended by A.R. Its. Dries Geudens is Research Assistant of the Fund for Scientific Research, Flanders (Belgium). Arno Kuijlaars is supported in part by FWO-Flanders projects G.0427.09 and G.0641.11, by KU Leuven research grant OT/08/33, by the Belgian Interuniversity Attraction Pole P06/02, and by a grant from the Ministry of Education and Science of Spain, project code MTM2005- 08648-C02-01.

Additional details

Created:
August 19, 2023
Modified:
October 23, 2023