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Gheissari, Reza; Newman, Charles M.; Stein, Daniel L.
Journal of statistical physics, 08/2018, Letnik: 172, Številka: 4Journal Article
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.
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