Critical Behavior of the Bethe Lattice Spin Glass

We present results of a rigorous analysis of the ± J Ising spin glass on the Bethe lattice with uncorrelated boundary conditions. We derive phase diagrams as functions of temperature vs. percentage of ferromagnetic bonds, and, when half of the bonds are ferromagnetic, temperature vs. external field. Critical exponents are also determined. Using bifurcation theory, we establish the existence of nontrivial distributions of single-site magnetizations within the low-temperature/small-field phase boundaries; these solutions reflect spin glass, ferromagnetic and magnetized spin glass behavior.