  We exhibit the multicritical phase structure of the loop gas model on a
random surface. The dense phase is reconsidered, with special attention paid to
the topological points $g=1/p$. This phase is complementary to the dilute and
higher multicritical phases in the sense that dense models contain the same
spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a
different set of boundary operators. This difference illuminates the well-known
$(p,q)$ asymmetry of the matrix chain models. Higher multicritical phases are
constructed, generalizing both Kazakov's multicritical models as well as the
known dilute phase models. They are quite likely related to multicritical
polymer theories recently considered independently by Saleur and Zamolodchikov.
Our results may be of help in defining such models on {\it flat} honeycomb
lattices; an unsolved problem in polymer theory. The phase boundaries
correspond again to ``topological'' points with $g=p/1$ integer, which we study
in some detail. Two qualitatively different types of critical points are
discovered for each such $g$. For the special point $g=2$ we demonstrate that
the dilute phase $O(-2)$ model does {\it not} correspond to the Parisi-Sourlas
model, a result likely to hold as well for the flat case. Instead it is proven
that the first {\it multicritical} $O(-2)$ point possesses the Parisi-Sourlas
supersymmetry.}
