  The model of p Ising spins coupled to 2d gravity, in the form of a sum over
planar phi-cubed 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 is performed and, using
a modified geodesic distance, various correlation functions are determined. The
two-point function has a diverging length scale associated with it. The
critical exponents are calculated and 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 with the transition to a branched polymer phase. Using a transfer
function we show that the two-point and spin-spin correlation functions decay
exponentially with distance. Finally, by assuming various scaling relations, we
make a prediction for the critical exponents at the transition between the
magnetized and branched polymer phases in the full model.
