  In recent years, a number of eccentric debris belts have been observed in
extrasolar systems. The most common explanation for their shape is the presence
of a nearby eccentric planetary companion. The gravitational perturbation from
such a companion would induce periodic eccentricity variations on the
planetesimals in the belt, with a range of precession frequencies. The overall
expected shape is an eccentric belt with a finite minimum width. However,
several observed eccentric debris discs have been found to exhibit a narrower
width than the theoretical expectation. In this paper, we study two mechanisms
that can produce this small width: (i) the protoplanetary disc can interact
with the planet and/or the planetesimals, slowly driving the eccentricity of
the former and damping the eccentricities of the latter; (ii) the companion
planet could have gained its eccentricity stochastically, through planet-planet
scatterings. We show that under appropriate conditions, both of these scenarios
offer a plausible way to reduce the minimum width of an eccentric belt exterior
to a perturbing planet. However, the effects of protoplanetary discs are
diminished at large separations (a > 10 au) due to the scarcity of gas and the
limited disc lifetime. These findings suggest that one can use the shape and
width of debris discs to shed light on the evolution of extrasolar systems,
constraining the protoplanetary disc properties and the prevalence of
planet-planet scatterings. Further observations of debris-harbouring systems
could confirm whether thin debris belts are a common occurrence, or the results
of rare initial conditions or evolutionary processes.
