The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\Gamma$ distributions. For such systems, the specific-heat exponent $\alpha$ does not equal the crossover exponent $\phi$ at random criticality. We derive a new relation between $\alpha$ and $\phi$, which invokes characteristics of the fixed distribution. The difference between $\alpha$ and $\phi$ is noted for $n$-vector models in $4-\epsilon$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\alpha$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\nu \simeq 1.06 \mathrm{and} 1.19$.

• Received 27 September 1983

DOI:https://doi.org/10.1103/PhysRevB.29.2630