We investigate thermodynamic properties of neural networks defined on a finite-dimensional lattice designed to store and retrieve patterns with structure. Our aim is to draw phase diagrams with axes of temperature and a parameter controlling the structure of patterns. Gauge symmetry is used to derive various exact or rigorous results on the properties of the system. These results put strong constraints on the possible phase diagrams. We also use Peierls arguments to prove the existence of a ferromagnetic phase and of a phase with finite overlap order in certain regions of the phase diagram. Our conclusion on the phase diagram is that, first, if the number of embedded patterns is smaller than a critical value, the system has in general three phases: a paramagnetic phase, a retrieval phase, and a ferromagnetic phase accompanied by finite overlap order. For larger numbers of embedded patterns, a ferromagnetic phase without overlap order appears in addition. The retrieval phase without ferromagnetic order may be replaced by a spin glass phase for large numbers of embedded patterns.
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