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

TL;DR

This paper derives an exact solution for the tail behavior of the free-energy distribution of an elastic string in a random potential, applicable to various boundary conditions and dimensions, with implications for growth models.

Contribution

It provides the first exact solution of the non-linear saddle-point equations for the optimal fluctuation problem in this context.

Findings

Exact form of the far-right tail of P(F) derived

Solution applicable for 0<d<2 in d-dimensional space

Results extend to the height distribution in KPZ growth models

Abstract

The free-energy distribution function of an elastic string in a quenched random potential, P(F), is investigated with the help of the optimal-fluctuation approach. The form of the far-right tail of P(F) is found by constructing the exact solution of the non-linear 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 imbeding 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.