  The transition in random fiber networks from two-dimensional to
three-dimensional planar structure driven by increasing coverage (total fiber
length per unit area) is studied with a deposition model. At low coverage the
network geometry depends on the scale-free product of fiber length and coverage
while at high coverage it depends on a scale-free combination of flexibility,
width and thickness of the fibers. With increasing coverage the roughness of
the free surface decouples from the substrate, faster when fibers are stiffer.
In the high coverage region roughness decreases exponentially with increasing
fiber flexibility.
