Speaker
Description
Employing the renormalization group method, we study the behavior of the explicitly anisotropic Hwa-Kardar sandpile model encompassed by an explicitly isotropic randomly moving medium. The motion of the medium is described by the stochastic Navier-Stokes equation with a random stirring force of a rather general form that includes, in particular, the overall shaking of the system and a non-local part with a power-law spectrum $k^{4-d-y}$ that describes, in the limiting case $y \to 4$, a turbulent pumping. The interplay between isotropic and anisotropic interactions results in a rich pattern of attractors of the RG flows in the space of the running coupling parameters, including fixed points, lines, and surfaces. The scaling dimensions corresponding to various asymptotic regimes are calculated perturbatively exactly (exactly to all orders of perturbation theory).
Moreover, the analysis of possible two-loop corrections suggests the existence of another nontrivial fixed point, not detectable within the one-loop approximation. If this point exists, it corresponds to a fully nontrivial regime where the Hwa-Kardar nonlinearity and advection by the turbulent fluid are simultaneously relevant. However, without practical two-loop calculations, the arguments supporting the existence of this fixed point are only preliminary and speculative. In particular, it is not clear whether the full-scale fixed point gives rise to a regular expansion in $y \sim \varepsilon$ or it cannot be treated within perturbation theory. This question remains for future study.
