### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 1009-1028 |

Number of pages | 20 |

Journal | Journal of Statistical Physics |

Volume | 172 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2018 |

### Fingerprint

### Keywords

- Constraint satisfaction
- Curie–Weiss model
- Dense random graph
- Minimum cut
- Random Ising model
- Zero-temperature dynamics

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*172*(4), 1009-1028. https://doi.org/10.1007/s10955-018-2087-9

**Zero-Temperature Dynamics in the Dilute Curie–Weiss Model.** / Gheissari, Reza; Newman, Charles; Stein, Daniel L.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 172, no. 4, pp. 1009-1028. https://doi.org/10.1007/s10955-018-2087-9

}

TY - JOUR

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

AU - Gheissari, Reza

AU - Newman, Charles

AU - Stein, Daniel L.

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

UR - http://www.scopus.com/inward/record.url?scp=85048885161&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048885161&partnerID=8YFLogxK

U2 - 10.1007/s10955-018-2087-9

DO - 10.1007/s10955-018-2087-9

M3 - Article

VL - 172

SP - 1009

EP - 1028

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -