  We construct a three-color Potts-like model for the graphene zigzag edge
reconstructed with Stone-Wales carbon rings, in order to study its thermal
equilibrium properties. We consider two cases which have different
ground-states: the edge with non-passivated dangling carbon bonds and the edge
fully passivated with hydrogen. We study the concentration of defects
perturbing the ground-state configuration as a function of the temperature. The
defect concentration is found to be exponentially dependent on the effective
parameters that describe the model at all temperatures. Moreover, we
analytically compute the domain size distribution of the defective domains and
conclude that it does not have fat-tails. In an appendix, we show how the
exchange parameters of the model can be estimated using density functional
theory results. Such equilibrium mechanisms place a lower bound on the
concentration of defects in zigzag edges, since the formation of such defects
is due to non-equilibrium kinetic mechanisms.
