New magnetic transition above the percolation threshold: Disordered Ising model on a Cayley tree

Magnetic phase transitions in a dilute Ising spin system on a bounded Cayley tree are dominated by the spins lying at the surface in the thermodynamic limit. An exact analytic treatment incorporating explicitly the spins at the surface of a site-disordered Cayley tree is developed for constant nearest-neighbor exchange interactions, and the existence of a new type of magnetic transition at low temperatures is demonstrated. This transition is characterized by the absence of spontaneous magnetization, by a divergence of the magnetic susceptibility at the percolation threshold $p_c$ as well as at ${\left(p_c\right)}^{\frac{1}{2}}$, and by a divergence of all higher-order zero-field correlation functions at $p_c$ and at higher concentrations forming a discrete set. The divergence of the susceptibility at $p_c$ marks the onset of ferromagnetic ordering in the central region of the Cayley tree. We also calculate explicitly the leading nonregular magnetic field dependence of the magnetization at high concentrations of spins, and show that the transition is effectively of infinite order. The possible relevance of our results to the transition in a diluted two-dimensional $\mathrm{xy}$ model is noted.