Scale-invariant quenched disorder and its stability criterion at random critical points

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 distributions. For such systems, the specific-heat exponent does not equal the crossover exponent at random criticality. We derive a new relation between and, which invokes characteristics of the fixed distribution. The difference between and is noted for n-vector models in 4- dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive 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 1.06 and 1.19.