Pavel Bleher, Alexander Its
Semiclassical Asymptotics of Orthogonal
Polynomials, Riemann-Hilbert Problem, and
Universality in the Matrix Model
(309K, TeX)
ABSTRACT. We derive semiclassical asymptotics for the
orthogonal polynomials on the line with the weight $\exp(-NV(z))$,
where $V(z)=\di{tz^2\over 2}+{gz^4\over 4},\;g>0,\;t<0$, is a
double-well quartic polynomial. Simultaneously we derive semiclassical
asymptotics for the recursive coefficients of the orthogonal
polynomials. The proof of the asymptotics is based on the
analysis of the appropriate matrix Riemann-Hilbert problem. As an
application of the semiclassical asymptotics, we prove the universality
of the local distribution of eigenvalues in the matrix model with
the double-well quartic interaction in the presence of two cuts.