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, overlineγ, related to γ by γ = {overlineγ}/{overlineγ-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 ( overlineγ, 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.