The Primal versus the Dual Ising Model

We represent the Ising model of statistical physics by Forney factor graphs in the primal and in the dual domains. By analogy with Kirchhoff's voltage and current laws, we show that in the primal Forney factor graph, the dependency among the variables is along the cycles, whereas in the dual Forney factor graph, the dependency is on the cutsets. In the primal (resp. dual) domain, dependent variables can be computed via their fundamental cycles (resp. fundamental cutsets). In each domain, we propose an importance sampling algorithm to estimate the partition function. In the primal domain, the proposal distribution is defined on a spanning tree, and computations are done on the cospanning tree. In contrast, in the dual domain, computations are done on a spanning tree of the model, and the proposal distribution is defined on the cospanning tree.