I am dealing with a lattice model that has the peculiar property that if I specify all the spins on the boundary, by local conservation laws, the whole lattice configuration (throughout the whole volume) is fixed. This trait is true for all $N\times N \times N$ lattices with this model. I am interested if in the thermodynamic limit this behavior holds and if this fixing prohibits phase transitions. Any ideas on these two matters?
Edit: These are things I think I can add. Since the internal variables are fixed by the boundary configurations, then for the partition function, there are bulk volume configurations where boundary spins are different but the internal spins are the same (compared to another configuration). Thus, I can define an equivalence between boundary configurations for two boundary configurations that yield the same interior. Then I only need to sum over equivalence class of surface configurations, and the model is somewhat 2D.