Roughening of the anharmonic Larkin model.

We study the roughening of d-dimensional directed elastic interfaces subject to quenched random forces. As in the Larkin model, random forces are considered constant in the displacement direction and uncorrelated in the perpendicular direction. The elastic energy density contains an harmonic part, proportional to (partial derivative(x)u)(2), and an anharmonic part, proportional to (partial derivative(x)u)(2n), where u is the displacement field and n > 1 an integer. By heuristic scaling arguments, we obtain the global roughness exponent zeta, the dynamic exponent z, and the harmonic to anharmonic crossover length scale, for arbitrary d and n, yielding an upper critical dimension d(c)(n) = 4n. We find a precise agreement with numerical calculations in d = 1. For the d = 1 case we observe, however, an anomalous "faceted" scaling, with the spectral roughness exponent zeta(s), satisfying zeta(s) > zeta > 1 for any finite n > 1, hence invalidating the usual single-exponent scaling for two-point correlation functions, and the small gradient approximation of the elastic energy density in the thermodynamic limit. We show that such d = 1 case is directly related to a family of Brownian functionals parameterized by n, ranging from the random-acceleration model for n = 1 to the Levy arcsine-law problem for n = infinity. Our results may be experimentally relevant for describing the roughening of nonlinear elastic interfaces in a Matheron-de Marsilly type of random flow.