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 \g=+1/2 or there exists a dual critical point with negative string susceptibility exponent, \g', related to \g by \g=\g'/(\g'-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 (\g',\g)=(-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.

18 pages, LaTeX file, two eps-figures

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• Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
• Rotational Symmetry Breaking in Multi-Matrix Models
• The Two-point Function of c=-2 Matter Coupled to 2D Quantum Gravity
• Hamiltonian Cycles on a Random Three-coordinate Lattice
• Spanning Forests on Random Planar Lattices
• Recursive sampling simulations of 3D gravity coupled to scalar fermions
• Phase transitions and random matrices
• Sums of Random Matrices and the Potts Model on Random Planar Maps
• Random Matrices, Boundaries and Branes