In this paper, we compute to high precision the roughness exponent ζ of a long-range elastic string, at the depinning threshold, in a random medium. Our numerical method exploits the analytic structure of the problem (“no-passing” theorem), but avoids direct simulation of the evolution equations. The roughness exponent has recently been studied by simulations, functional renormalization-group calculations, and by experiments (fracture of solids, liquid meniscus in ${}^4\mathrm{He}).\mathrm{}$ Our result $\zeta =0.388\mathrm{}\pm 0.002$ is significantly larger than what was stated in previous simulations, which were consistent with a one-loop renormalization-group calculation. Furthermore, the data are incompatible with the experimental results for crack propagation in solids and for a ${}^4\mathrm{He}$ contact line on a rough substrate. This implies that the experiments cannot be described by pure harmonic long-range elasticity in the quasistatic limit.

• Received 27 July 2001

DOI:https://doi.org/10.1103/PhysRevE.65.025101