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.