MiS Preprint Repository

We derive probabilistic limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume-Emery-Griffiths model. These probabilistic limit theorems consist of scaling limits for the total spin and moderate deviation principles (MDPs) for the total spin. The model under study is defined by a probability distribution that depends on the parameters $n, \beta,$ and $K,$ which represent, respectively, the number of spins, the inverse temperature, and the interaction strength. The intricate structure of the phase transitions is revealed by the existence of 18 scaling limits and 18 MDPs for the total spin. These limit results are obtained as $(\beta,K)$ converges along appropriate sequences $(\beta_n,K_n)$ to points belonging to various subsets of the phase diagram, which include a curve of second-order points and a tricritical point. The forms of the limiting densities in the scaling limits and of the rate functions in the MDPs reflect the influence of one or more sets that lie in neighborhoods of the critical points and the tricritical point.

Of all the scaling limits, the structure of those near the tricritical point is by far the most complex, exhibiting new types of critical behavior when observed in a limit-theorem phase diagram in the space of the two parameters that parametrize the scaling limits. The scaling limits and the MDPs are derived via a unified method based on two components: analyzing the Taylor expansions of a function $G_{\beta,K}$ whose minimum value equals the canonical free energy and using a large deviation principle with rate function $G_{\beta,K}$ to control various error terms arising in the proofs of the limit theorems.