  The low lying spectrum of the massless overlap Dirac operator coupled to
abelian fields in three dimensions with three different measures are shown to
exhibit two phases: a strong coupling gapped phase and a weak coupling gapless
phase. The vanishing of the gap from the strong coupling side with a Maxwell
and a conformal measure is governed by a Gaussian exponent. Contrary to this
result, the vanishing of the gap from the strong coupling side with a compact
Thirring measure is not consistent with a Gaussian exponent. The low lying
spectrum with a non-compact Thirring measure does not exhibit a simple
non-monotonic behavior as a function of the lattice size on the weak coupling
side. Our combined analysis suggests exploring the possibility of a strongly
coupled continuum theory starting from a compact lattice Thirring model where a
compact U(1) gauge field with a single link action is coupled to even number of
flavors of massless overlap Dirac fermions.
