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, [ital w][sup 2], is found to scale as [ital P]([ital w][sup 2])=[l angle][ital w][sup 2][r angle][sup [minus]1][Phi]([ital w][sup 2]/[l angle][ital w][sup 2][r angle]) where [l angle][ital w][sup 2][r angle] is the average of [ital w][sup 2]. We calculate the scaling function [Phi]([ital x]) exactly and compare it both to exact enumerations for a discrete-slope surface evolution model and to [Phi]'s obtained in Monte Carlo simulations of equilibrium and driven interfaces of chemically reacting systems.