### Abstract

We consider the complexity of random ferromagnetic landscapes on the hypercube {±1}N given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph G(N, p). Previous results had shown that, with high probability as N→ ∞, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as N→ ∞. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.

Original language | English (US) |
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Journal | Journal of Statistical Physics |

DOIs | |

State | Accepted/In press - Jan 1 2020 |

### Keywords

- Complex landscapes
- Constrained optimization
- Disordered ferromagnets
- Dynamics in dilute Curie–Weiss models
- Landscape search algorithms
- Local MINCUT
- Local energy minima
- Mean-field ferromagnets
- Metastable traps
- Quenched disorder

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*. https://doi.org/10.1007/s10955-019-02480-4