CORRELATION-FUNCTIONS IN THE MULTIPLE ISING-MODEL COUPLED TO GRAVITY

The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar phi(3) graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations i s performed and, using a modified geodesic distance, various correlati on functions are determined. The two-point function has a diverging le ngth scale associated with it. The critical exponents are calculated a nd it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model , which also has a diverging geometric correlation length associated w ith the transition to a branched polymer phase. Using a transfer funct ion we show that the two-point and spin-spin correlation functions dec ay exponentially with distance. Finally, by assuming various scaling r elations, we make a prediction for the critical exponents at the trans ition between the magnetized and branched polymer phases in the full m odel.