Exact Results for Quenched Bond Randomness at Criticality

We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the $q$-state Potts model, we show that a line of renormalization group fixed points interpolates from weak to strong randomness as $q-2$ grows from small to large values. This theory exhibits a $q$-independent sector, and allows at the same time for a correlation length exponent which keeps the Ising value and continuously varying magnetization exponent and effective central charge. These findings appear to solve long-standing numerical and theoretical puzzles, and to illustrate the peculiarities which may characterize the conformal field theories of random fixed points.

Figure 1

Scattering processes corresponding to the amplitudes $S_0$, $S_1$, $S_2$, $S_3$, $S_4$, $S_5$, and $S_6$, in that order. Different latin indices correspond to different replicas, and different greek letters for the same replica correspond to different colors.