Large Deviations on a Cayley Tree I: Rate Functions

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions the ferromagnetic phase transition takes place at the critical temperature $T_c=\frac{6\sqrt{2}}{5}J$, where $J$ is the interaction strength. For any temperature the equilibrium magnetization, $m_n$, tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization $r_n=n^{3/2}m_n$, where $n$ is the number of generations in the Cayley tree. Below $T_c$, the equilibrium values of the order parameter are given by $$\rho^* = \pm\frac{2\pi} {(\sqrt{2}-1)^2} \sqrt{1-\frac{T}{T_c}}.$$ There is one more notable temperature, $T_{\rm p}$, in the model. Below that temperature the influence of homogeneous boundary field penetrates throughout the tree. We call $T_{\rm p}$ the penetration temperature, and it is given by $$T_{\rm p}= \frac{J} {W_{\rm Cayley} (3/2)} \left(1-\frac{1}{\sqrt{2}} \left( \frac{h}{2J} \right)^2 \right).$$ The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.