  Many protoplanetary discs are self-gravitating early in their lives. If they
fragment under their own gravity, they form bound gaseous clumps which may
evolve to become giant planets. Today, the fraction of discs that undergo
fragmentation, and the frequency of conditions that may lead to giant planet
formation via gravitational instability, is still unknown.
  We perform a population synthesis of discs from formation to dispersal. In
varying the infall radius, we study the relationship of the early disc size
with fragmentation. Furthermore, we investigate how stellar accretion heating
affects the fragmentation fraction.
  We find that discs fragment only if they become sufficiently large early in
their lives. This size depends sensitively on where mass is added to the discs
during the collapse of their parent molecular cloud core. By choosing
intermediate infall locations leading to a synthetic disc size distribution
that is in agreement with the observed one, we find a fragmentation fraction
between 0.1 and 11 %, depending on the efficiency of stellar accretion heating
of the discs.
  We conclude that the early disc size is mainly determined by the infall
location during the collapse of the molecular cloud core and controls the
population-wide frequency of fragmentation. Stellar accretion heating plays an
important role for fragmentation and must be studied further. Our work is an
observationally-informed step towards a prediction of the frequency of giant
planet formation by gravitational instability. Upcoming observations and
theoretical studies will deepen our understanding of the formation and early
evolution of discs, eventually allowing to understand how infall, disc
morphology, giant planet formation via gravitational instability, and the
observed extrasolar planet population are linked.
