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The Kardar-Parisi-Zhang equation with spatially correlated noise: a unified picture from nonperturbative renormalization group

We investigate the scaling regimes of the Kardar-Parisi-Zhang equation in the presence of spatially correlated noise with power law decay $D(p) \sim p^{-2ρ}$ in Fourier space, using a nonperturbative renormalization group approach. We determine the full phase diagram of the system as a function of $ρ$ and the dimension $d$. In addition to the weak-coupling part of the diagram, which agrees with the results from Refs. [Europhys. Lett. 47, 14 (1999), Eur. Phys. J. B 9, 491 (1999)], we find the two fixed points describing the short-range (SR) and long-range (LR) dominated strong-coupling phases. In contrast with a suggestion in the references cited above, we show that, for all values of $ρ$, there exists a unique strong-coupling SR fixed point that can be continuously followed as a function of $d$. We show in particular that the existence and the behavior of the LR fixed point do not provide any hint for 4 being the upper critical dimension of the KPZ equation with SR noise.