  We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on
the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a
double-well quartic polynomial, in the limit when n, N \to \infty. We assume
that \epsilon \le (n/N) \le \lambda_{cr} - \epsilon for some \epsilon > 0,
where \lambda_{cr} is the critical value which separates orthogonal polynomials
with two cuts from the ones with one cut. Simultaneously we derive
semiclassical asymptotics for the recursive coefficients of the orthogonal
polynomials, and we show that these coefficients form a cycle of period two
which drifts slowly with the change of the ratio n/N. The proof of the
semiclassical asymptotics is based on the methods of the theory of integrable
systems and on the analysis of the appropriate matrix Riemann-Hilbert problem.
As an application of the semiclassical asymptotics of the orthogonal
polynomials, 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.
