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March 2021 Tightness and tails of the maximum in 3D Ising interfaces
Reza Gheissari, Eyal Lubetzky
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Ann. Probab. 49(2): 732-792 (March 2021). DOI: 10.1214/20-AOP1459


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/logncβ in probability as n.

Here, we show that the laws of the centered maxima (MnE[Mn])n1 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.


We are grateful to an anonymous referee for valuable comments. E.L. was supported in part by NSF Grant DMS-1812095.


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Reza Gheissari. Eyal Lubetzky. "Tightness and tails of the maximum in 3D Ising interfaces." Ann. Probab. 49 (2) 732 - 792, March 2021.


Received: 1 September 2019; Revised: 1 April 2020; Published: March 2021
First available in Project Euclid: 17 March 2021

Digital Object Identifier: 10.1214/20-AOP1459

Primary: 60K35 , 82B20 , 82B24

Keywords: 3D Ising model , low temperature interface , maximum of random surface

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 2 • March 2021
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