Multicritical Phases of the O(n) Model on a Random Lattice

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.}