  We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+
>...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z))
dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient
$\kappa_n$ is found to satisfy a difference-differential (spatially discrete)
equation which is further proved to approach a third order differential
equation by double scaling. The third order differential equation is equivalent
to the Painlev\'e II equation. The leading coefficient and second leading
coefficient of $\phi_n(z)$ can be expressed asymptotically in terms of the
Painlev\'e II function.
