# Lattice model completely constrained by boundary data

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.

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Interesting problem! Some more details might be useful. It sounds like you might get different results for the thermodynamic limit depending on what boundary conditions you choose, e.g. periodic b.c. allow you freedom that fixed-spin b.c. do not. There's at least one example of this sort of thing that I can find: arxiv.org/abs/cond-mat/0004250 - and there's a little discussion about b.c. and the thermodynamic limit in Baxter's Exactly Solved Models book, which is free online: physics.anu.edu.au/theophys/baxter_book.php – AJK May 17 '13 at 2:38
I think that the thermodynamic limit is either ill-defined or trivial in this case, as it depends crucially on boundary conditions. – Ikiperu May 20 '13 at 17:18
I'd like to see more details of your model. That such a thing can exist at all is surprising and interesting! – Nathaniel May 21 '13 at 1:42