Width distribution for random-walk interfaces

Roughening of a one-dimensional interface is studied under the assumption that the interface configurations are continuous, periodic random walks. The distribution of the square of the width of interface, ${\mathit{w}}^2$, is found to scale as P(${\mathit{w}}^2$)=〈${\mathit{w}}^2$ ${\mathrm{〉}}^{\mathrm{-}1}$Φ(${\mathit{w}}^2$/〈${\mathit{w}}^2$〉) where 〈${\mathit{w}}^2$〉 is the average of ${\mathit{w}}^2$. We calculate the scaling function Φ(x) exactly and compare it both to exact enumerations for a discrete-slope surface evolution model and to Φ’s obtained in Monte Carlo simulations of equilibrium and driven interfaces of chemically reacting systems.