  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 $(\partial_x u)^2$, and an anharmonic part, proportional to
$(\partial_x u)^{2n}$, 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 L\'evy arcsine-law problem for $n = \infty$. Our results may be
experimentally relevant for describing the roughening of non-linear elastic
interfaces in a Matheron-de Marsilly type of random flow.

Disordered Systems and Neural Networks