Universal and non-universal tails of distribution functions in the directed polymer and KPZ problems

The optimal fluctuation approach is applied to study the most distant (non-universal) tails of the free-energy distribution function P(F) for an elastic string (of a large but finite length L) interacting with a quenched random potential. A further modification of this approach is proposed which takes into account the renormalization effects and allows one to study the most close (universal) parts of the tails. The problem is analyzed for different dimensions of a space in which the polymer is imbedded. In terms the stochastic growth problem, the same distribution function describes the distribution of heights in the regime of a non-stationary growth in a situation when an interface starts to grow from a flat configuration.