The long-time, large-distance behaviour of a randomly walking particle on a random rough surface in a uniform gravitational field is studied by means of the field-theoretic renormalization group (RG). The random walk is governed by the Fokker–Planck equation for the particle’s probability distribution function, while the surface is described by the conserved Kardar–Parisi–Zhang (CKPZ) model due to Sun, Guo and Grant [Phys. Rev. A 40 6763 (1989)]. The corresponding field-theoretic model is logarithmic at \(d = 2\) and is shown to be multiplicatively renormalizable. The standard RG analysis based on the minimal subtraction scheme (where the ultraviolet divergences have the form of the poles in \(\varepsilon =2-d\) ) reveals no infrared (IR) attractive fixed points. However, it shows that a certain IR irrelevant (in the sense of Wilson) term (composite operator) in the corresponding De Dominicis–Janssen action functional, necessarily omitted in that scheme, is in fact inevitably needed to exhaustively describe the IR behaviour of the relevant Green’s functions. Thus, we use a non-conventional formulation of the renormalization scheme in fixed d dimensions, in which that operator is included from the very beginning into the action functional and is treated on the equal footing with the other terms. That scheme reveals at least one nontrivial IR attractive fixed point, elusive in the standard RG approach. The resulting asymptotic scaling expressions appear rather cumbersome, as they simultaneously involve two different critical dimensions of time/frequency (so-called weak scaling) and describe several different asymptotic sub-domains of the whole IR asymptotic region.
