Explicit solution of the optimal fluctuation problem for an elastic string in a random medium

The free-energy distribution function of an elastic string in a quenched random potential, P-L(F), is investigated with the help of the optimal fluctuation approach. The form of the far-right tail of P-L(F) is found by constructing the exact solution of the nonlinear saddle-point equations describing the asymptotic form of the optimal fluctuation. The solution of the problem is obtained for two different types of boundary conditions and for an arbitrary dimension of the imbedding space 1+d with d from the interval 0 < d < 2. The results are also applicable for the description of the far-left tail of the height distribution function in the stochastic growth problem described by the d-dimensional Kardar-Parisi-Zhang equation.