Ising spin system on a Cayley tree: Correlation decomposition and phase transition

For N spins, σᵢ=±1, i∈1,2,,N, interacting via nearest-neighbor ferromagnetic Ising interaction -Jσᵢσⱼ on a Cayley tree with branching number B, it is shown that any even-spin correlation function 〈σ_i₁σ_i₂⋯σ_i2K〉 decomposes into a product 〈σ_j₁σ_j₂〉⋯〈σ_j_2K-1σ_j2K〉 of two-spin correlation functions 〈σ_jₚσ_jₚ₊₁〉=[tanh(JkBT)]^d(jₚ, jₚ₊₁), where d(jₚ, jₚ₊₁) is the number of bonds on the unique self-avoiding path connecting σ_jₚ and σ_jₚ₊₁. This generalizes to $B>1$ the known decomposition for an Ising chain (a Cayley tree having $B=1$). The decomposition theorem leads to upper and lower bounds for the zero-field susceptibility, and these bounds become infinite for temperatures T≤T₂ and are finite for T>T₂ where Btanh²(JkBT₂)=1. An upper bound is also given for the fourth cumulant of the magnetization. That bound becomes (negatively) infinite for T<T₄ where B³tanh⁴(JkBT₄)=1. The above exact considerations are consistent with recent results of other authors and provide elementary insight regarding the cumulant divergences and long-range correlation of subsets of surface spins.