  We consider random deposition of debris or blocks on a line, with block sizes
following a rigorous hierarchy: the linear size equals $1/\lambda^n$ in
generation $n$, in terms of a rescaling factor $\lambda$. Without interactions
between the blocks, this model is described by a logarithmic fractal, studied
previously, which is characterized by a constant increment of the length, area
or volume upon proliferation. We study to what extent the logarithmic
fractality survives, if each block is equipped with an Ising (pseudo-)spin
$s=\pm 1$ and the interactions between those spins are switched on (ranging
from antiferromagnetic to ferromagnetic). It turns out that the dependence of
the surface topology on the interaction sign and strength is not trivial. For
instance, deep in the ferromagnetic regime, our numerical experiments and
analytical results reveal a sharp crossover from a Euclidean transient,
consisting of aggregated domains of aligned spins, to an asymptotic logarithmic
fractal growth. In contrast, deep into the antiferromagnetic regime the surface
roughness is important and is shown analytically to be controlled by vacancies
induced by frustrated spins. Finally, in the weak interaction regime, we
demonstrate that the non-interacting model is extremal in the sense that the
effect of the introduction of interactions is only quadratic in the magnetic
coupling strength. In all regimes, we demonstrate the adequacy of a mean-field
approximation whenever vacancies are rare. In sum, the logarithmic fractal
character is robust with respect to the introduction of spatial correlations in
the hierarchical deposition process.
