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We numerically study the geometry of a driven elastic string at itssample-dependent depinning threshold in random-periodic media. We find that theanisotropic finite-size scaling of the average square width $\bar{w^2}$ and ofits associated probability distribution are both controlled by the ratio$k=M/L^{\zeta_{\mathrm{dep}}}$, where $\zeta_{\mathrm{dep}}$ is therandom-manifold depinning roughness exponent, $L$ is the longitudinal size ofthe string and $M$ the transverse periodicity of the random medium. Therescaled average square width $\bar{w^2}/L^{2\zeta_{\mathrm{dep}}}$ displays anon-trivial single minimum for a finite value of $k$. We show that the initialdecrease for small $k$ reflects the crossover at $k \sim 1$ from therandom-periodic to the random-manifold roughness. The increase for very large$k$ implies that the increasingly rare critical configurations, accompanyingthe crossover to Gumbel critical-force statistics, display anomalous roughnessproperties: a transverse-periodicity scaling in spite that $\bar{w^2} \ll M$,and subleading corrections to the standard random-manifold longitudinal-sizescaling. Our results are relevant to understanding the dimensional crossoverfrom interface to particle depinning.

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