Simplest nonequilibrium phase transition into an absorbing state

We study in further detail particle models displaying a boundary-induced absorbing state phase transition. 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 nontrivial 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.