Distributions of absolute central moments for random walk surfaces

We study periodic Brownian paths, wrapped around the surface of a cylinder. One characteristic of such a path is its width square, $w^2$, defined as its variance. Though the average of $w^2$ over all possible paths is well known, its full distribution function was investigated only recently. Generalising $w^2$ to $w^{(N)}$, defined as the $N$-th power of the {\it magnitude} of the deviations of the path from its mean, we show that the distribution functions of these also scale and obtain the asymptotic behaviour for both large and small $w^{(N)}$.