Dynamics of Particle Deposition on a Disordered Substrate: I. Near-   Equilibrium Behavior
  A growth model which describes the deposition of particles (or the growth of a rigid crystal) on a disordered substrate is investigated. The dynamic renormalization group is applied to the stochastic growth equation using the Martin, Sigga, and Rose formalism. The periodic potential and the quenched disorder, upon averaging, are combined into a single term in the generating functional. Changing the temperature (or the inherent noise of the deposition process) two different regimes with a transition between them at $T_{sr}$, are found: for $T>T_{sr}$ this term is irrelevant and the surface has the scaling properties of a surface growing on a flat substrate in the rough phase. The height-height correlations behave as $C(L,\tau)\sim \ln [L f(\tau/L^2)]$. While the linear response mobility is finite in this phase it does vanish as $(T-T_{sr})^{1.78}$ when $T\rightarrow T_{sr}^+$. For $T<T_{sr}$ there is a line of fixed-point for the coupling constant. The surface is super-rough: the equilibrium correlation functions behave as $(\ln L)^2$ while their short time dependence is $(\ln \tau) ^2$ with a temperature dependent dynamic exponent $z=2[1+1.78(1-T/T_{sr})]$. While the linear response mobility vanishes on large length scales, its scale-dependence leads to a non-linear response. For a small applied force $F$ the average velocity of the surface $v$ behaves as $v \sim F^{1+\zeta}$. To first order $\zeta=1.78(1-T/T_{sr})$. At the transition, $v\sim F/(1+C|\ln (F)|)^{1.78}$ and the crossover to the behavior to $T<T_{sr}$ is analyzed. These results also apply to two-dimensional vortex glasses with a parallel magnetic field. 
Yan-Chr Tsai and Yonathan Shapir