Exact results for the Ising model on a small-world network

Small-world networks provide an interesting framework for studying the interplay between regular and random graphs, where links are located in a regular and random way, respectively. On one hand, the random links make the model to obey some kind of mean-field behavior. On the other hand, the links of the regular lattice make the system to retain some related non trivial correlations. The coexistence of these two features in general prevent a closed analytical treatment. Here we consider a one-dimensional small-world Ising model and derive analytically its equation of state, critical point, critical behavior, and critical correlations. Despite being one of the simplest small-world models, our exact and intuitive analysis reveals some intriguing properties.