  We study in further detail particle models displaying a boundary-induced
absorbing state phase transition [Phys. Rev. E. {\bf 65}, 046104 (2002) and
Phys. Rev. Lett. {\bf 100}, 165701 (2008)] . These are one-dimensional systems
consisting of a single site (the boundary) where creation and annihilation of
particles occur and a bulk where particles move diffusively. We study different
versions of these models, and confirm that, except for one exactly solvable
bosonic variant exhibiting a discontinuous transition and trivial exponents,
all the others display non-trivial behavior, with critical exponents differing
from their mean-field values, representing a universality class. Finally, the
relation of these systems with a $(0+1)$-dimensional non-Markovian process is
discussed.
