  Ring-like features have been observed in several debris discs. Outside the
main ring, while some systems exhibit smooth surface brightness profiles (SB)
that fall off roughly as r**-3.5, others display large luminosity drops at the
ring's outer edge and steeper radial SB profiles. We seek to understand this
diversity of outer edge profiles under the ``natural'' collisional evolution of
the system, without invoking external agents such as planets or gas. We use a
statistical code to follow the evolution of a collisional population, ranging
from dust grains (submitted to radiation pressure) to planetesimals and
initially confined within a belt (the 'birth ring'). The system typically
evolves toward a "standard" steady state, with no sharp edge and SB \propto
r**-3.5 outside the birth ring. Deviations from this standard profile, in the
form of a sharp outer edge and a steeper fall-off, occur only when two
parameters take their extreme values: 1) When the birth ring is so massive that
it becomes radially optically thick for the smallest grains. However, the
required disc mass is here probably too high to be realistic. 2) When the
dynamical excitation of the dust-producing planetesimals is so low (<e> <0.01)
that the smallest grains, which otherwise dominate the total optical depth, are
preferentially depleted. This low-excitation case, although possibly not
generic, cannot be ruled out by observations. Our "standard" profile provides a
satisfactory explanation for a large group of debris discs with outer edges and
SB falling as r**-3.5. Systems with sharper outer edges, barring other
confining agents, could still be explained by ``natural'' collisional evolution
if their dynamical excitation is very low. We show that such a dynamically-cold
case provides a satisfactory fit for HR4796A
