Percolation, clusters, and phase transitions in spin models

The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability p depending on the NN coupling constant J and the temperature T, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability p=1-exp(-2JkT). The BCPM has the Ising critical temperature and exponents, including ν,ν^',η,β,α,α^', and γ (perhaps also γ^'). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.