Tightness and tails of the maximum in 3D Ising interfaces

Reza Gheissari, Eyal Lubetzky

Research output: Contribution to journalArticlepeer-review


Consider the 3D Ising model on a box of side length n with minus boundary conditions above the xy-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height Mn of this interface: for every β large, Mn/log n cβ in probability as n ∞ Here, we show that the laws of the centered maxima (Mn — (Formula presented)[Mn])n >1 are uniformly tight. Moreover, even though this sequence does not converge, we prove that it has uniform upper and lower Gumbel tails (exponential right tails and doubly exponential left tails). Key to the proof is a sharp (up to O(1) precision) understanding of the surface large deviations. This includes, in particular, the shape of a pillar that reaches near-maximum height, even at its base, where the interactions with neighboring pillars are dominant.

Original languageEnglish (US)
Pages (from-to)732-792
Number of pages61
JournalAnnals of Probability
Issue number2
StatePublished - Mar 2021


  • 3D ising model
  • low temperature interface
  • maximum of random surface

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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