Zero-Temperature Dynamics in the Dilute Curie–Weiss Model

We consider the Ising model on a dense Erdős–Rényi random graph, G(N, p) , with p> 0 fixed—equivalently, a disordered Curie–Weiss Ising model with 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 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.