We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states, q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent $\eta_n$ for several moments, n, of the correlation functions, including typical (n -> 0), average (n=1) and others. In addition, for q=8, we find that there is only a single correlation length exponent describing the correlation length away from criticality. This is numerically very close to the pure Ising value of unity.
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