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 quench from $T_i \rightarrow \infty$ to $T \leqslant T_\mathrm{s}$ , where $T_\mathrm{s}$ 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 unitary structures while the system goes to the equilibrium when these are cyclic, no matter the coordination number (z) of the particularly considered lattice.