Universal width distributions in non-Markovian Gaussian processes

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^α. 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+δ]. 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 δ, but we show that it is universal (independent of boundary conditions) in the small-window limit. We compute it directly for all values of α, using, for α<3, an asymptotic expansion formula that we derive. For α> 3, the limiting width distribution is independent of α. 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

15 pages

References in corpus (13)

• Universal Fluctuations in Correlated Systems
• Nonequilibrium Dynamics of Interfaces and Lines
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Magnetic fluctuations in the classical XY model: the origin of an exponential tail in a complex system
• Width distribution of contact lines on a disordered substrate
• Width Distributions and the Upper Critical Dimension of KPZ Interfaces
• Crack roughness and avalanche precursors in the random fuse model
• Spatial Persistence of Fluctuating Interfaces
• Roughness distributions for 1/f^alpha signals
• Universal interface width distributions at the depinning threshold
• Higher correlations, universal distributions and finite size scaling in the field theory of depinning
• Large scale numerical simulations of "ultrametric" long-range depinning
• Search for universal roughness distributions in a critical interface model

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• Maximum relative height of elastic interfaces in random media
• Bath induced phase transition in a Luttinger liquid
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• Anisotropic finite-size scaling of an elastic string at the depinning threshold in a random-periodic medium
• Statistics of zero crossings in rough interfaces with fractional elasticity