  Using an exact mapping to disordered Coulomb gases, we introduce a novel
method to study two dimensional Dirac fermions with quenched disorder in two
dimensions which allows to treat non perturbative freezing phenomena. For
purely random gauge disorder it is known that the exact zero energy eigenstate
exhibits a freezing-like transition at a threshold value of disorder
$\sigma=\sigma_{th}=2$. Here we compute the dynamical exponent $z$ which
characterizes the critical behaviour of the density of states around zero
energy, and find that it also exhibits a phase transition. Specifically, we
find that $\rho(E=0 + i \epsilon) \sim \epsilon^{2/z-1}$ (and $\rho(E) \sim
E^{2/z-1}$) with $z=1 + \sigma$ for $\sigma < 2$ and $z=\sqrt{8 \sigma} - 1$
for $\sigma > 2$. For a finite system size $L<\epsilon^{-1/z}$ we find large
sample to sample fluctuations with a typical $\rho_{\epsilon}(0) \sim L^{z-2}$.
Adding a scalar random potential of small variance $\delta$, as in the
corresponding quantum Hall system, yields a finite noncritical $\rho(0) \sim
\delta^{\alpha}$ whose scaling exponent $\alpha$ exhibits two transitions, one
at $\sigma_{th}/4$ and the other at $\sigma_{th}$. These transitions are shown
to be related to the one of a directed polymer on a Cayley tree with random
signs (or complex) Boltzmann weights. Some observations are made for the strong
disorder regime relevant to describe transport in the quantum Hall system.
