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 gamma = + 1/2 or there exists a dual critical point with negative string susceptibility exponent, <(gamma)over tilde>, related to gamma by gamma = <(gamma)over tilde>/<(gamma)over tilde>. 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 (<(gamma)over tilde>, gamma) = (- 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.