Correlation length-exponent relation for the two-dimensional random Ising model

We consider the two-dimensional (2d) random Ising model on a diagonal strip of the square lattice, where the bonds take two values, $J_1>J_2$, with equal probability. Using an iterative method, based on a successive application of the star-triangle transformation, we have determined at the bulk critical temperature the correlation length along the strip, ${\xi }_L$, for different widths of the strip, $L\le 21$. The ratio of the two lengths, ${\xi }_L\mathrm{/}L=A$, is found to approach the universal value, $A=2\mathrm{/}\pi$ for large $L$, independent of the dilution parameter, $J_1\mathrm{/}{J}_2$. With our method we have demonstrated with high numerical precision, that the surface correlation function of the 2d dilute Ising model is self-averaging, in the critical point conformally coovariant and the corresponding decay exponent is ${\eta }_{\parallel }=1$.