Large deviations and a phase transition in the Block Spin Potts models

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, hence the name block spin Potts model. The values a spin can take are called colors. 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 and appropriately chosen interaction strength 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.

Energy landscape of the two-component Curie-Weiss-Potts model with three spins

“…This has been investigated in recent literature. For example, [12] presented the law of large numbers and large deviation principle for the empirical magnetizations for the multi-component Curie-Weiss-Potts model. The corresponding central limit theorem and moderate deviation principle have been investigated in [10], and the central limit theorem for the joint distribution for empirical magnetization in the two-component Curie-Weiss-Potts model has been analyzed in [16] by inspecting the limiting behavior of the free energy.…”