Fast counting and sampling for ferromagnetic two-spin systems

We introduce two new models equivalent to ferromagnetic two-spin systems: a weighted subgraph model and a random cluster type model. Using these new connections, we obtain an efficient sampling algorithm and a new randomised algorithm that efficiently approximates the partition function of ferromagnetic two-spin systems in certain parameter regimes. No efficient sampling algorithms are known before in this regime, and our new estimation algorithm runs in near-quadratic time for bounded degree graphs and in polynomial time for general graphs, improving upon the previous algorithm of Guo, Liu, and Lu (2020).