  A generic class of scalar active matter, characterized at the mean field
level by the diffusivity vanishing above some threshold density, was recently
introduced [Golestanian R 2019 Phys. Rev. E 100 010601(R)]. In the presence of
harmonic confinement, such 'diffusivity edge' was shown to lead to condensation
in the ground state, with the associated transition exhibiting formal
similarities with Bose-Einstein condensation (BEC). In this work, the effect of
a diffusivity edge is addressed in a periodic potential in arbitrary
dimensions, where the system exhibits coexistence between many condensates.
Using a generalized thermodynamic description of the system, it is found that
the overall phenomenology of BEC holds even for finite energy barriers
separating each neighbouring pair of condensates. Shallow potentials are shown
to quantitatively affect the transition, and introduce non-universality in the
values of the scaling exponents.
