  We report a transfer matrix study of the random bond $q-$state Potts model in
the vicinity of the Ising model $q=2$. We draw attention to a precise
determination of magnetic scaling dimensions in order to compare with
perturbative results. Universality is checked by the computation of the
spin-spin correlation function decay exponent obtained with different types of
probability distributions of the coupling strengths. Our numerical data,
compared to perturbative results for the second moment of the correlation
function, obtained with both replica symmetry and replica symmetry breaking
schemes, are conclusively in favour of the replica symmetric calculations. The
multifractal behaviour of higher moments as well as that of typical correlation
functions are also investigated and a comparison is made with the perturbative
expansions. Finally, the shape of the correlation function probability
distribution is analyzed.
