  The electrodynamical response of the edge of a compressible Quantum Hall
system affects tunneling into the edge. Using the composite Fermi liquid
theory, we derive an effective action for the edge modes interacting with
tunneling charge. This action generalizes the chiral Luttinger liquid theory of
the Quantum Hall edge to compressible systems in which transport is
characterized by a finite Hall angle. In addition to the standard terms, the
action contains a dissipative term. The tunneling exponent is calculated as a
function of the filling fraction for several models, including short-range and
long-range Coulomb interaction. We find that tunneling exponents are robust and
to a large extent insensitive to the particular model.
  We discuss recent tunneling measurements in the overgrown cleaved edge
systems, and demonstrate that the profile of charge density near the edge is
very sensitive to the parameters of the system. In general, the density is
non-monotonic, and can deviate from the bulk value by up to 30%. Implications
for the correspondence to the chiral Luttinger edge theories are discussed.
