Phase structure of the O( n) model on a random lattice for n > 2

Abstract

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.