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

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 and of its associated probability distribution are both controlled by the ratio , where is the random-manifold depinning roughness exponent, is the longitudinal size of the string and the transverse periodicity of the random medium. The rescaled average square width displays a non-trivial single minimum for a finite value of . We show that the initial decrease for small reflects the crossover at from the random-periodic to the random-manifold roughness. The increase for very large 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 , 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. Received: 20 October 2010,  Accepted: 1 December 2010; Edited by: A. Vindigni; Reviewed by: A. A. Fedorenko, CNRS-Lab. de Physique, ENS de Lyon, France; DOI: 10.4279/PIP.020008