Some cluster-size and percolation problems for interacting spins

The problem of cluster-size distribution and percolation for interacting spins on a regular lattice is briefly discussed. Exact solutions are given for Bethe lattices and other more complex branching media. It is found that the critical behavior is not changed with respect to the noninteracting case. For a ferromagnetic interaction the critical density $p_c$ has been found to be always less than the corresponding critical density in the random distribution. Moreover, at zero external magnetic field $p_c$ has been always ≤ 1/2, which means that an infinite cluster of overturned spins appears before the Curie temperature is reached. The pair connectedness is also calculated for the simple Bethe lattices and it is found to satisfy homogeneity conditions.