LAUSR

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions a ferromagnetic phase transition takes place at the critical temperature $$T_\mathrm{c} =\frac{6\sqrt{2}}{5}J$$Tc=625J, where J is the interaction strength. For any temperature the equilibrium magnetization, $$m_n$$mn, tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization $$r_n=n^{3/2}m_n$$rn=n3/2mn, where n is the number of generations in the Cayley tree. Below $$T_\mathrm{c}$$Tc, the equilibrium values of the order parameter are given by $$\pm \rho ^*$$±ρ∗, where $$\begin{aligned} \rho ^*=\frac{2\pi }{(\sqrt{2}-1)^2}\sqrt{1-\frac{T}{T_\mathrm{c}}}. \end{aligned}$$ρ∗=2π(2-1)21-TTc.One more notable temperature in the model is the penetration temperature $$\begin{aligned} T_\mathrm{p}=\frac{J}{W_\mathrm{Cayley}(3/2)}\left( 1-\frac{1}{\sqrt{2}}\left( \frac{h}{2J}\right) ^2\right) . \end{aligned}$$Tp=JWCayley(3/2)1-12h2J2.Below $$T_\mathrm{p}$$Tp the influence of homogeneous boundary field of magnitude h penetrates throughout the tree. The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.