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

For $N$ spins, ${\sigma }_i=\pm 1$, $i\in \{1,2,\dots ,N\}$, interacting via nearest-neighbor ferromagnetic Ising interaction $-J{\sigma }_i{\sigma }_j$ on a Cayley tree with branching number $B$, it is shown that any even-spin correlation function $〈{\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{2K}}〉$ decomposes into a product $〈{\sigma }_{j_1}{\sigma }_{j_2}〉\cdots 〈{\sigma }_{j_{2K-1}}{\sigma }_{j_{2K}}〉$ of two-spin correlation functions $〈{\sigma }_{j_p}{\sigma }_{j_{p+1\mathrm{}}}〉={\left[tanh\right(\frac{J}{k_{B}T}\left)\right]}^{d(j_p, j_{p+1\mathrm{}})}$, where $d(j_p, j_{p+1\mathrm{}})$ is the number of bonds on the unique self-avoiding path connecting ${\sigma }_{j_p}$ and ${\sigma }_{j_{p+1\mathrm{}}}$. This generalizes to $B>\mathrm{}1\mathrm{}$ the known decomposition for an Ising chain (a Cayley tree having $B=1\mathrm{}$). The decomposition theorem leads to upper and lower bounds for the zero-field susceptibility, and these bounds become infinite for temperatures $T\mathit{\le }{T}_2$ and are finite for $T>\mathrm{}{T}_2$ where $Btan{\mathrm{h}}^2\left(\frac{J}{k_B{T}_2}\right)=1\mathrm{}$. An upper bound is also given for the fourth cumulant of the magnetization. That bound becomes (negatively) infinite for $T<\mathrm{}{T}_4$ where $B^{3}tan{\mathrm{h}}^4\left(\frac{J}{k_B{T}_4}\right)=1\mathrm{}$. 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.