Random-field Ising model on a Bethe lattice

The ground state of the random-field Ising ferromagnet on a Bethe lattice is found. With decreasing random-field strength an infinite series of spin-flip transitions precedes the onset of ferromagnetism. The $T=0\mathrm{}$ critical behavior is not mean-field-like. At finite temperatures a series of Griffiths singularities precedes the phase transition. As $T\to 0$, the Griffiths singularities terminate at the spin-flip transitions. The $T\ne 0$ critical behavior is argued to be mean-field-like.