Distributions of absolute central moments for random walk surfaces

TL;DR

This paper analyzes the distribution of absolute central moments for periodic Brownian paths on a cylinder, generalizing the width squared to higher moments and exploring their asymptotic behaviors.

Contribution

It introduces the distribution functions of higher-order absolute central moments for wrapped Brownian paths and derives their asymptotic properties.

Findings

Distribution functions scale with the moments.

Asymptotic behavior characterized for large and small moments.

Generalization from width squared to higher moments.

Abstract

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 $w^{(N)}$.