Coniglio-Klein mapping in the metastable region

We explicitly calculate the connectivity in the percolation problem defined by the Coniglio-Klein mapping of the Ising model on the Bethe lattice. We study the relation between thermal correlations and connectivity in the [formula presented] region where the system is metastable, with the aim of interpreting the mean-field spinodal line with a percolation line of the same sort. We find that the extension of the mapping to the metastable region is characterized by a nontrivial feature, i.e., the simultaneous presence of two infinite percolating clusters of opposite spin. This feature destroys the usual equivalence between correlation and connectivity. A different relation between thermal and percolative quantities, which reduces to the know relation in the stable region, can be obtained if the cross correlation between the two infinite clusters is taken into account.