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, w², is found to scale as P(w²)=〈w²{〉}^{{{-}}1}Φ({w}²$/〈${w}²$〉) where 〈${w}²$〉 is the average of ${w}²$. 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.