We study the combined effect of growth (material deposition from above) and

nearest-neighbor entropic and force-dipole interactions in a stochastically perturbed

system of N line defects (steps) on a vicinal crystal surface in 1+1 dimensions.

First, we formulate a general model of conservative white noise, and we derive sim-

plified formulas for the terrace width distribution (TWD) and pair correlations,

particularly the covariance matrix of terrace widths, in the limit N → ∞ for small

step fluctuations. Second, we apply our formalism to two specific noise models

which stem, respectively, from: (i) the fluctuation-dissipation theorem for diffusion

of adsorbed atoms; and (ii) the phenomenological consideration of deposition-flux-

induced asymmetric attachment and detachment of atoms at step edges. We discuss

implications of our analysis, particularly the narrowing of the TWD with the de-

position flux, connection of noise structure to terrace width correlations, behavior

of these correlations in the macroscopic limit, and comparison of our perturbation

results to a known mean field approach.