  The interplay between the critical fluctuations and the sample geometry is
investigated numerically using thin random-field ferromagnets exhibiting the
field-driven magnetisation reversal on the hysteresis loop. The system is
studied along the theoretical critical line in the plane of random-field
disorder and thickness. The thickness is varied to consider samples of various
geometry between a two-dimensional plane and a complete three-dimensional
lattice with an open boundary in the direction of the growing thickness. We
perform a multi-fractal analysis of the Barkhausen noise signals and scaling of
the critical avalanches of the domain wall motion. Our results reveal that, for
sufficiently small thickness, the sample geometry profoundly affects the
dynamics by modifying the spectral segments that represent small fluctuations
and promoting the time-scale dependent multi-fractality. Meanwhile, the
avalanche distributions display two distinct power-law regions, in contrast to
those in the two-dimensional limit, and the average avalanche shapes are
asymmetric. With increasing thickness, the scaling characteristics and the
multi-fractal spectrum in thicker samples gradually approach the hysteresis
loop criticality in three-dimensional systems. Thin ferromagnetic films are
growing in importance technologically, and our results illustrate some new
features of the domain wall dynamics induced by magnetisation reversal in these
systems.
