Dynamics of Surface Roughening with Quenched Disorder

We study the dynamical exponent $z$ for the directed percolation depinning (DPD) class of models for surface roughening in the presence of quenched disorder. We argue that $z$ for $d+1\mathrm{}$ dimensions is equal to the exponent $d_{\min }$ characterizing the shortest path between two sites in an isotropic percolation cluster in $d$ dimensions. To test the argument, we perform simulations and calculate $z$ for DPD, and $d_{\min }$ for percolation, from $d=1\mathrm{}$ to $d=6\mathrm{}$.