### Abstract

We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)^{-p}, p > 2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value J _{c}, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to J _{c}, the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as J → J_{c} ^{-}. (In d = 3 stripes mean slabs, not columns.) Here we rigorously prove that, if we normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio e _{0}(J)/e _{S}(J) tends to 1 as J → J_{c} ^{-}, with e _{S}(J) being the energy per site of the optimal periodic striped/slabbed state and e _{0}(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e _{0}(J)-e _{S}(J) at small but positive J _{c}-J, and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: namely, if one looks at a randomly chosen window, of suitable size ℓ (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability.

Original language | English (US) |
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Pages (from-to) | 333-350 |

Number of pages | 18 |

Journal | Communications In Mathematical Physics |

Volume | 331 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2014 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*331*(1), 333-350. https://doi.org/10.1007/s00220-014-1923-2