Abstract
We study the roughening of -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 , and an anharmonic part, proportional to , where is the displacement field and an integer. By heuristic scaling arguments, we obtain the global roughness exponent , the dynamic exponent , and the harmonic to anharmonic crossover length scale, for arbitrary and , yielding an upper critical dimension . We find a precise agreement with numerical calculations in . For the case we observe, however, an anomalous “faceted” scaling, with the spectral roughness exponent satisfying for any finite , 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 case is directly related to a family of Brownian functionals parameterized by , ranging from the random-acceleration model for to the Lévy arcsine-law problem for . Our results may be experimentally relevant for describing the roughening of nonlinear elastic interfaces in a Matheron-de Marsilly type of random flow.
2 More- Received 28 December 2018
DOI:https://doi.org/10.1103/PhysRevE.99.032105
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