Abstract
We introduce and analyze a generalization of the blocks spin Ising (Curie-Weiss) models that were discussed in a number of recent articles. In these block spin models each spin in one of s blocks can take one of a finite number of q≥3 values or colors, hence the name block spin Potts model. We prove a large deviation principle for the percentage of spins of a certain color in a certain block. These values are represented in an s×q matrix. We show that for uniform block sizes there is a phase transition. In some regime the only equilibrium is the uniform distribution of all colors in all blocks, while in other parameter regimes there is one predominant color, and this is the same color with the same frequency for all blocks. Finally, we establish log-Sobolev-type inequalities for the block spin Potts model.
Highlights
Mean-field models as the Curie–Weiss model are approximations of lattice models
We prove a large deviation principle for the percentage of spins of a certain color in a certain block
We establish log-Sobolev-type inequalities for the block spin Potts model
Summary
Mean-field models as the Curie–Weiss model are approximations of lattice models. They often show qualitatively interesting results (see [13] for a survey). Large deviations in the block spin Potts model models). Let us mention at this point that, while we were finishing the current manuscript we learned that in [29] the author studies a very similar model: Here the number of blocks is restricted to two, but they may be of different size. Extending his results, we prove a large deviation principle, are able to locate the minima of the rate functions and show logarithmic Sobolev inequalities
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