Exact critical behavior of a random bond two-dimensional Ising model

A 2D Ising model in which the bonds K fluctuate randomly about ${\mathrm{K}}_{\mathrm{c}}$, the critical value of the pure system, is considered. The ensemble average of the square of the two-point function, <<${\mathrm{s}}_0$ ${\mathrm{s}}_{\mathrm{R}}$ ${\mathrm{〉}}^2$ ${\mathrm{〉}}_{\mathrm{Av}}$, is shown to decay as (lnR${)}^{1/4}$ ${\mathrm{R}}^{\mathrm{-}1/2}$ aat the critical point. This implies that <<${\mathrm{s}}_0$ ${\mathrm{s}}_{\mathrm{R}}$ ${>>}_{\mathrm{Av}}$ is bounded above by (lnR${)}^{1/8}$ ${\mathrm{R}}^{\mathrm{-}1/4}$ in disagreement with the exp [-(ln lnR${)}^2$] decay law found by Dotsenko and Dotsenko by a different method. On the other hand, the present calculation reproduces their specific-heat singularity C∼ln‖lnτ‖ (τ=K-${\mathrm{K}}_{\mathrm{c}}$).