TY - JOUR

T1 - Zero-Temperature Dynamics in the Dilute Curie–Weiss Model

AU - Gheissari, Reza

AU - Newman, Charles M.

AU - Stein, Daniel L.

N1 - Funding Information:
Acknowledgements The authors would like to thank the anonymous referee for useful comments and suggestions. The research of R.G. and C.M.N. was supported in part by US-NSF Grant DMS-1507019.
Funding Information:
Funding was provided by National Science Foundation (Grant No. 1207678).
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Constraint satisfaction

KW - Curie–Weiss model

KW - Dense random graph

KW - Minimum cut

KW - Random Ising model

KW - Zero-temperature dynamics

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U2 - 10.1007/s10955-018-2087-9

DO - 10.1007/s10955-018-2087-9

M3 - Article

AN - SCOPUS:85048885161

VL - 172

SP - 1009

EP - 1028

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -