Local Minima in Disordered Mean-Field Ferromagnets

Eric Yilun Song, Reza Gheissari, Charles M. Newman, Daniel L. Stein

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)576-596
Number of pages21
JournalJournal of Statistical Physics
Issue number1-6
StatePublished - Sep 1 2020


  • 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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