Anisotropic finite-size scaling of an elastic string at the depinning threshold in a random-periodic medium

Abstract

We numerically study the geometry of a driven elastic string at its sample-dependent depinning threshold in random-periodic media. We find that the anisotropic finite-size scaling of the average square width w2 and of its associated probability distribution are both controlled by the ratio k = M/Lζdep , where ζdep is the random-manifold depinning roughness exponent, L is the longitudinal size of the string and M the transverse periodicity of the random medium. The rescaled average square width w2/L2ζdep displays a non-trivial single minimum for a finite value of k. We show that the initial decrease for small k reflects the crossover at k ∼ 1 from the random-periodic to the random-manifold roughness. The increase for very large k implies that the increasingly rare critical configurations, accompanying the crossover to Gumbel critical-force statistics, display anomalous roughness properties: a transverse-periodicity scaling in spite that w2 M, and subleading corrections to the standard random-manifold longitudinal-size scaling. Our results are relevant to understanding the dimensional crossover from interface to particle depinning.