Non-local meta-conformal invariance in diffusion-limited erosion

Abstract

The non-stationary relaxation and physical ageing in the diffusion-limited erosion process ({\sc dle}) is studied through the exact solution of its Langevin equation, in $d$ spatial dimensions. The dynamical exponent $z=1$, the growth exponent $\beta=\max(0,(1-d)/2)$ and the ageing exponents $a=b=d-1$ and $\lambda_C=\lambda_R=d$ are found. In $d=1$ spatial dimension, a new representation of the meta-conformal Lie algebra, isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$, acts as a dynamical symmetry of the noise-averaged {\sc dle} Langevin equation. Its infinitesimal generators are non-local in space. The exact form of the full time-space dependence of the two-time response function of {\sc dle} is reproduced for $d=1$ from this symmetry. The relationship to the terrace-step-kink model of vicinal surfaces is discussed.