A two-term truncation of the multiple Ising model coupled to 2-D gravity

We consider a model of p independent Ising spins on a dynamical planar phi(3) graph. Truncating the free energy to two terms yields an exactly solvable model that has a third-order phase transition from a pure gravity region (gamma(str) = -1/2) to a tree-like region (gamma(str) = 1/2), with gamma(str) = 1/3 on the critical line. We are able to make an order of magnitude estimate of the value of p above which there exists a branched polymer (i.e. tree-like) phase in the full model, that is, p similar to 13-23, which corresponds to a central charge of c similar to 6-12.