Interface energy in random systems

We study the interface energy $\sigma$ as a function of disorder in two-dimensional Ising-type systems at $T=0\mathrm{}$ for the percolation and frustration models. Our approach consists in calculating this energy for long strips of varying widths by a random sampling method, then extrapolating the results. In the weak-disorder limit, very wide strips may be studied and accurate values are obtained for the first-order correction to the interface energy in both models. Only moderate widths (up to $N=9-10\mathrm{}$) can be studied in the general situation and we use finite-size scaling to analyze the data in the region of the threshold, where $\sigma$ vanishes with a critical exponent $v$. For percolation, we obtain $\frac{v}{\nu }=0.98\mathrm{}\pm 0.05$, where $\nu$ is the correlation-length exponent, in agreement with Deutscher and Rappaport's proposal that $v=\nu$ exactly. The analysis of the results is more difficult in the case of frustration, because size effects are important and the scaling region is not reached for $N=9\mathrm{}$. Our data show that Monte Carlo results for $\sigma$ are unreliable and that much care is necessary to reach firm conclusions on the frustration threshold $x_c$ or the exponents $\nu$ and $v$.