Freezing vs. equilibration dynamics in the Potts model

We study the quench dynamics of the $q$ Potts model on different bi/tri-dimensional lattice topologies. In particular we are interested in instantaneous quenches from $T_i\to \mathrm{\infty }$ to $T\le T_s$, where $T_s$ is the (pseudo)-spinodal temperature. The goal is to explain why, in the large-$q$ limit, the low-temperature dynamics freezes on some lattices while, on others, the equilibrium configuration is easily reached. The cubic ($3d$) and the triangular ($2d$) lattices are analysed in detail. We show that the dynamics blocks when lattices have acyclic $\mathit{\text{unitary}}\mathit{\text{structures}}$ while the system goes to the equilibrium when these are cyclic, no matter the coordination number ($z$) of the particular considered lattice.