From trees to galaxies: The Potts model on a random surface Original Research Article

The matrix model of random surfaces with c = ∞ has recently been solved and found to be identical to a random surface coupled to a q-states Potts model with q = ∞. 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 = ∞ 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 qc, 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.