  The Ising antiferromagnet is an important statistical physics model with
close connections to the {\sc Max Cut} problem. Combining spatial mixing
arguments with the method of moments and the interpolation method, we pinpoint
the replica symmetry breaking phase transition predicted by physicists.
Additionally, we rigorously establish upper bounds on the {\sc Max Cut} of
random regular graphs predicted by Zdeborov\'a and Boettcher [Journal of
Statistical Mechanics 2010]. As an application we prove that the
information-theoretic threshold of the disassortative stochastic block model on
random regular graphs coincides with the Kesten-Stigum bound.
