  We study the effects of quenched disorder and a dissipative Coulomb
interaction on an anyon gas in a periodic potential undergoing a quantum phase
transition. We use a $(2+1)$d low-energy effective description that involves
$N_f = 1$ Dirac fermion coupled to a $U(1)$ Chern-Simons gauge field at level
$(\theta - 1/2)$. When $\theta = 1/2$ the anyons are free Dirac fermions that
exhibit an integer quantum Hall transition; when $\theta = 1$ the anyons are
bosons undergoing a superconductor-insulator transition in the universality
class of the 3d XY model. Using the large $N_f$ approximation we perform a
renormalization group analysis. The dissipative Coulomb interaction allows for
two classes of IR stable fixed points: those with a finite, nonzero Coulomb
coupling and dynamical critical exponent $z = 1$ and those with an effectively
infinite Coulomb coupling and $1 < z < 2$. We find the Coulomb interaction to
be an irrelevant perturbation of the clean fixed point for any $\theta$. At
$\theta = 1/2$ the clean fixed point is stable to charge-conjugation preserving
(random mass) disorder, while a line of diffusive fixed points obtains when the
product of charge-conjugation and time-reversal symmetries is preserved. At
$\theta = 1$ we find a finite disorder fixed point with unbroken
charge-conjugation symmetry whether or not the Coulomb interaction is present.
Other cases result in runaway flows. We comment on the relation of our results
to other theoretical studies and the relevancy to experiment.
