Open Access
2023 Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor
Reza Gheissari, Eyal Lubetzky
Author Affiliations +
Electron. J. Probab. 28: 1-44 (2023). DOI: 10.1214/23-EJP987

Abstract

We study the interface of the Ising model in a box of side-length n in Z3 at low temperature 1β under Dobrushin’s boundary conditions, conditioned to stay in a half-space above height h (a hard floor). Without this conditioning, Dobrushin showed in 1972 that typically most of the interface is flat at height 0. With the floor, for small h, the model is expected to exhibit entropic repulsion, where the typical height of the interface lifts off of 0. Detailed understanding of the SOS model—a more tractable height function approximation of 3D Ising—due to Caputo et al., suggests that there is a single integer value hnclogn of the floor height, delineating the transition between rigidity at height 0 and entropic repulsion.

We identify an explicit hn=(c+o(1))logn such that, for the typical Ising interface above a hard floor at h, all but an ε(β)-fraction of the sites are propelled to be above height 0 if h<hn1, whereas all but an ε(β)-fraction of the sites remain at height 0 if hhn. Further, c is such that the typical height of the unconditional maximum is (2c+o(1))logn; this confirms scaling predictions from the SOS approximation.

Acknowledgments

R.G. thanks the Miller Institute for Basic Research in Science for its support. The research of E.L. was supported in part by NSF grants DMS-1812095 and DMS-2054833.

Citation

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Reza Gheissari. Eyal Lubetzky. "Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor." Electron. J. Probab. 28 1 - 44, 2023. https://doi.org/10.1214/23-EJP987

Information

Received: 21 December 2022; Accepted: 28 June 2023; Published: 2023
First available in Project Euclid: 18 July 2023

MathSciNet: MR4616514
zbMATH: 07733576
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP987

Subjects:
Primary: 60K35 , 82B20

Keywords: Entropic repulsion , Interface , Ising model , Random surface

Vol.28 • 2023
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