Mathematics – Classical Analysis and ODEs
Scientific paper
2010-07-19
Nonlinearity 24 (2011), 951--993
Mathematics
Classical Analysis and ODEs
60 pages, 9 figures
Scientific paper
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.
Duits Maurice
Geudens Dries
Kuijlaars Arno B. J.
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