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

作者：Duits Maurice^*; Geudens Dries; Kuijlaars Arno B J

来源：Nonlinearity, 2011, 24(3): 951-993.

DOI：10.1088/0951-7715/24/3/012

We consider the two sequences of biorthogonal polynomials (p(k,n))(k=0)(infinity) and (q(k,n))(k=0)(infinity) 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 -> 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 behaviour for a certain negative value of t.
We also prove a general result about the interlacing of zeros of biorthogonal polynomials.

• 出版日期2011-3