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arXiv · 1007.3137

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

Maurice Duits·Dries Geudens·Arno B. J. Kuijlaars
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Abstract

We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and (q_{k,n})_k 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 tends to infinity. 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 behavior for a certain negative value of t. We also prove a general result about the interlacing of zeros of biorthogonal polynomials.

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Maurice Duits, Dries Geudens, Arno B. J. Kuijlaars. 2010-07-19. A vector equilibrium problem for the two-matrix model in the quartic/quadratic case. https://doi.org/10.1088/0951-7715%2F24%2F3%2F012

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