Phase Structure of the O(n) Model on a Random Lattice for N > 2

We show that coarse graining arguments invented for the analysis of multi-spin systems on a randomly triangulated surface apply also to the O(n) model on a random lattice. These arguments imply that if the model has a critical point with diverging string susceptibility, then either γ = + 1 2 or there exists a dual critical point with negative string susceptibility exponent, ˜ γ, related to γ by γ = ˜ γ ˜ γ−1. Exploiting the exact solution of the O(n) model on a random lattice we show that both situations are realized for n > 2 and that the possible dual pairs of string susceptibility exponents are given by (˜ γ, γ) = − 1 m , 1 m+1 , m = 2, 3,. . .. We also show that at the critical points with positive string susceptibility exponent the average number of loops on the surface diverges while the average length of a single loop stays finite.