The phase diagram of an Ising model on a polymerized random surface

We construct a random surface model with a string susceptibility exponent γ = 1/4 by taking an Ising model on a random surface and introducing an additional degree of freedom which amounts to allowing certain outgrowths on the surfaces. Fine-tuning the Ising temperature and the weight factor for outgrowths we find a triple point where γ = 1/4. At this point magnetized and nonmagnetized gravity phases meet a branched polymer phase.