Anomalous Height Fluctuation Width in Crossover from Random to Coherent Surface Growths

We study an anomalous behavior of the height fluctuation width in the crossover from random to coherent growths of surface for a stochastic model. In the model, random numbers are assigned on perimeter sites of surface, representing pinning strengths of disordered media. At each time, surface is advanced at the site having minimum pinning strength in a random subset of system rather than having global minimum. The subset is composed of a randomly selected site and its $(\ell-1)$ neighbors. The height fluctuation width $W^2(L;\ell)$ exhibits the non-monotonic behavior with $\ell$ and it has a minimum at $\ell^*$. It is found numerically that $\ell^*$ scales as $\ell^*\sim L^{0.59}$, and the height fluctuation width at that minimum, $W^2(L;\ell^*)$, scales as $\sim L^{0.85}$ in 1+1 dimensions. It is found that the subset-size $\ell^*(L)$ is the characteristic size of the crossover from the random surface growth in the KPZ universality, to the coherent surface growth in the directed percolation universality.