  In this paper we provide new analytic results on two-dimensional $q$-Potts
models ($q \geq 2$) in the presence of bond disorder correlations which decay
algebraically with distance with exponent $a$. In particular, our results are
valid for the long-range bond disordered Ising model ($q=2$). We implement a
renormalization group perturbative approach based on conformal perturbation
theory. We extend to the long-range case the RG scheme used in [V. Dotsenko,
Nucl. Phys. B 455 701 23] for the short-range disorder. Our approach is based
on a $2$-loop order double expansion in the positive parameters $(2-a)$ and
$(q-2)$. We will show that the Weinrib-Halperin conjecture for the long-range
thermal exponent can be violated for a non-Gaussian disorder. We compute the
central charges of the long-range fixed points finding a very good agreement
with numerical measurements.
