FROM TREES TO GALAXIES - THE POTTS-MODEL ON A RANDOM SURFACE

The matrix model of random surfaces with c = infinity has recently been solved and found to be identical to a random surface coupled to a q-states Ports model with q = infinity. The mean field-like solution exhibits a novel type of tree structure. The natural question is, down to which-if any-finite values of c and q does this behavior persist? In this work we develop, for the Potts model, an expansion in the fluctuations about the q = infinity mean field solution. In the lowest-cubic-non-trivial order in this expansion the corrections to mean field theory can be given a nice interpretation in terms of structures (trees and ''galaxies'') of spin clusters. When q drops below a finite q(c), the galaxies overwhelm the trees at all temperatures, thus suppressing mean field behavior. Thereafter the phase diagram resembles that of the Ising model, q = 2.