PHASE-STRUCTURE OF THE O(N) MODEL ON A RANDOM LATTICE FOR N-GREATER-THAN-2

We show that coarse graining arguments invented for the analysis of mu lti-spin systems on a randomly triangulated surface apply also to the O(n) model on a random lattice. These arguments imply that if the mode l has a critical point with diverging string susceptibility, then eith er gamma = + 1/2 or there exists a dual critical point with negative s tring susceptibility exponent, <(gamma)over tilde>, related to gamma b y gamma = <(gamma)over tilde>/<(gamma)over tilde>. Exploiting the exac t solution of the O(n) model on a random lattice we show that both sit uations are realized for n > 2 and that the possible dual pairs of str ing susceptibility exponents are given by (<(gamma)over tilde>, gamma) = (- 1/m, 1/m+1), m = 2,3,... We also show that at the critical point s with positive string susceptibility exponent the average number of l oops on the surface diverges while the average length of a single loop stays finite.