We study the effects of turbulent mixing on the random growth of an interface in the problem of the deposition of a substance on a substrate. The growth is modeled by the well-known Kardar–Parisi– Zhang model. The turbulent advecting velocity field is modeled by the Kraichnan rapid-change ensemble: Gaussian statistics with the correlation function hvvi ∝ δ(t − t ′ )k −d−ξ , where k is the wave number and ξ is a free parameter, 0 <ξ <2. We study the effects of the fluid compressibility. Using the field theory renormalization group, we show that depending on the relation between the exponent ξ and the spatial dimension d, the system manifests different types of large-scale, long-time asymptotic behavior associated with four possible fixed points of the renormalization group equations. In addition to the known regimes (ordinary diffusion, the ordinary growth process, and a passively advected scalar field), we establish the existence of a new nonequilibrium universality class. We calculate the fixed-point coord