We study a block spin mean-field Ising model. For the vector of block magnetizations we prove Large Deviation Principles and Central Limit Theorems under general assumptions for the block interaction matrix. Using the exchangeable pair approach of Stein's method we are also able to establish a speed of convergence for the Central Limit Theorem for the vector of block magnetizations in the high temperature regime.

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.

Abstract. We analyze the so called Swapping Algorithm, a parallel version of the well-known Metropolis-Hastings algorithm, on the mean-field version of the BlumeEmery-Griffiths model in statistical mechanics. This model has two parameters and depending on their choice, the model exhibits either a first, or a second order phase transition. In agreement with a conjecture by Bhatnagar and Randall we find that the Swapping Algorithm mixes rapidly in presence of a second order phase transition, while becoming slow when the phase transition is first order.

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We investigate the behavior of the free energy of the p-spin interaction variant of the SK-model for high temperatures. We admit arbitrary distributions of the interactions given that their distribution is symmetric around the origin and some exponential moment is finite. We show that there is a critical temperatureβ depending on p such that for β <β the free energy of the p-spin interaction model has normally distributed fluctuations.