Correlated site-bond percolation for multicomponent spin models

Abstract

Abstract The problem of correlated site-bond percolation is solved exactly for classical multicomponent spin models on a Cayley tree. It is shown, that the Coniglio-Klein result for the critical percolation probability p c at a phase transition can be generalised to read p c = |λ k |, where λ k is the critical value of an eigenvalue of the interaction matrix exp — β E ( i,j ). Some exact phase diagrams illustrating this point are derived for the Ising and Potts models. For this latter model, the presence of “high-field” phase transitions makes the phase diagrams rather complicated. The relation p c = |λ k | is also discussed for systems on real lattices.