From Trees to Galaxies: The Potts Model on a Random Surface

TL;DR

This paper investigates the behavior of the Potts model on random surfaces, revealing how finite q values influence the transition from tree-like to galaxy-like structures in spin clusters, affecting the phase diagram.

Contribution

It develops an expansion around the q=inf. mean field solution for the Potts model on random surfaces, elucidating the transition from tree to galaxy structures in spin clusters.

Findings

Galaxies dominate over trees below a critical q_c.

Mean field behavior is suppressed when q < q_c.

Phase diagram resembles the Ising model for q=2.

Abstract

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