Abstract
In this chapter, we illustrate the aggregate path coupling method for models with one-dimensional macroscopic quantities. The model chosen is the mean-field Blume-Capel model where the relevant macroscopic quantity is the one-dimensional magnetization S n ∕n. The mean-field Blume-Capel model is ideally suited for the analysis of the relationship between the thermodynamic equilibrium behavior and mixing times due to its intricate phase transition structure. Specifically, as discussed in Subsection 3.2.2, the phase diagram of the model includes a curve at which the model undergoes a continuous phase transition, a curve where the model undergoes a first-order phase transition, and a tricritical point which separates the two curves.
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