Competing Criticality of Short- and Infinite-Range Interactions on the Cayley Tree

The Ising model, with equivalent-neighbor and nearest-neighbor interactions of Cayley tree connectivity, is solved exactly. Breaking translational symmetry by turning on the Cayley interactions is analogous to lowering spatial dimensionality in Bravais lattices. A range of classical criticality, a point of logarithmic corrections, a range of continuously varying power-law singularities, and a point of exponential singularity are successively encountered.