Maximum relative height of elastic interfaces in random media

TL;DR

This paper investigates the distribution of maximum relative height in elastic interfaces within random media, comparing exact numerical results with Gaussian mode approximations, and finds boundary conditions significantly influence the distribution.

Contribution

It provides an exact analytical description of the right tail of the MRH distribution and compares different configurations and boundary conditions in elastic interfaces.

Findings

Corrections from Gaussian modes are generally small.

In the large size limit, the MRH distribution for the ground state in the random-periodic class approaches the Airy distribution.

MRH distributions are sensitive to boundary condition changes.

Abstract

The distribution of the maximal relative height (MRH) of self-affine one-dimensional elastic interfaces in a random potential is studied. We analyze the ground state configuration at zero driving force, and the critical configuration exactly at the depinning threshold, both for the random-manifold and random-periodic universality classes. These configurations are sampled by exact numerical methods, and their MRH distributions are compared with those with the same roughness exponent and boundary conditions, but produced by independent Fourier modes with normally distributed amplitudes. Using Pickands' theorem we derive an exact analytical description for the right tail of the latter. After properly rescaling the MRH distributions we find that corrections from the Gaussian independent modes approximation are in general small, as previously found for the average width distribution of depinning configurations. In the large size limit all corrections are finite except for the ground-state in the random-periodic class whose MRH distribution becomes, for periodic boundary conditions, indistinguishable from the Airy distribution. We find that the MRH distributions are, in general, sensitive to changes of boundary conditions.