  Universal behavior is a typical emergent feature of critical systems. A
paramount model of the non-equilibrium critical behavior is the directed bond
percolation process that exhibits an active- to-absorbing state phase
transition in the vicinity of a percolation threshold. Fluctuations of the
ambient environment might affect or destroy the universality properties
completely. In this work we assume that the random environment can be described
by means of compressible velocity fluctu- ations. Using field-theoretic models
and renormalization group methods we investigate large-scale and long-time
behavior. Altogether eleven universality classes are found, out of which four
are stable in the infrared limit and thus macroscopically accessible. In
contrast to the model without veloc- ity fluctuations a possible candidate for
a realistic three-dimensional case, a regime with relevant short-range noise,
is identified. Depending on the dimensionality of space and the structure of
the turbulent flow we calculate critical exponents of the directed percolation
process. In the limit of the purely transversal velocity field random force
critical exponents comply with the incompressible results obtained by previous
authors. We have found intriguing non-universal behavior related to the mutual
effect of compressibility and advection.
