Zero-temperature dynamics in the dilute Curie-Weiss model

We consider the Ising model on a dense Erd\H{o}s--R\'enyi random graph, $\mathcal G(N,p)$, with $p>0$ fixed---equivalently, a disordered Curie--Weiss Ising model with $\mbox{Ber}(p)$ couplings---at zero temperature. The disorder may induce local energy minima in addition to the two uniform ground states. In this paper we prove that, starting from a typical initial configuration, the zero-temperature dynamics avoids all such local minima and absorbs into a predetermined one of the two uniform ground states. We relate this to the local MINCUT problem on dense random graphs; namely with high probability, the greedy search for a local MINCUT of $\mathcal G(N,p)$ with $p>0$ fixed, started from a uniform random partition, fails to find a non-trivial cut. In contrast, in the disordered Curie--Weiss model with heavy-tailed couplings, we demonstrate that zero-temperature dynamics has positive probability of absorbing in a random local minimum different from the two homogenous ground states.