Phase Structure of the O(n) Model on a Random Lattice for n>2

TL;DR

This paper analyzes the phase structure of the O(n) model on a random lattice for n>2, revealing the existence of dual critical points with specific string susceptibility exponents and divergent loop behavior at criticality.

Contribution

It applies coarse graining arguments to the O(n) model on random lattices and identifies the dual pairs of string susceptibility exponents for n>2.

Findings

Both critical points with positive and negative string susceptibility exponents are realized for n>2.

Dual pairs of exponents are given by (-1/m, 1/(m+1)) for m=2,3,...

At critical points with positive exponents, the number of loops diverges while loop length remains finite.

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.