Universal width distributions in non-Markovian Gaussian processes

TL;DR

This paper investigates how boundary conditions affect the width distribution of non-Markovian Gaussian processes and demonstrates a universal behavior in the small-window limit, providing exact formulas and asymptotic expansions.

Contribution

It derives the universal width distribution for non-Markovian Gaussian processes across all spectral exponents and provides exact and asymptotic formulas, advancing understanding of boundary effects.

Findings

Width distribution depends on boundary conditions for finite windows.

Universal width distribution emerges in the small-window limit.

Exact formulas are provided for all spectral exponents.

Abstract

We study the influence of boundary conditions on self-affine random functions u(t) in the interval t/L \in [0,1], with independent Gaussian Fourier modes of variance ~ 1/q^{\alpha}. We consider the probability distribution of the mean square width of u(t) taken over the whole interval or in a window t/L \in [x, x+\delta]. Its characteristic function can be expressed in terms of the spectrum of an infinite matrix. This distribution strongly depends on the boundary conditions of u(t) for finite \delta, but we show that it is universal (independent of boundary conditions) in the small-window limit. We compute it directly for all values of \alpha, using, for \alpha<3, an asymptotic expansion formula that we derive. For \alpha > 3, the limiting width distribution is independent of \alpha. It corresponds to an infinite matrix with a single non-zero eigenvalue. We give the exact expression for the width distribution in this case. Our analysis facilitates the estimation of the roughness exponent from experimental data, in cases where the standard extrapolation method cannot be used