On the rigidity of sloped height functions in d≥ 3 and non-crossing surfaces
Abstract
We consider integer-valued ∇ϕ height functions, with general convex interactions, placed on a slope. Sheffield (2003) conjectured that for all slopes, such height functions are localized in dimensions d≥3, in the sense of having tight fluctuations in finite volume, and admitting infinite-volume limits. We establish this conjecture when there are two coordinates on which the slope vector is zero and on which the interactions are even and low temperature. In this setting, we also prove the uniqueness, in the appropriate sense, of the infinite-volume limit and classify its extremal components. We further study the structural properties of the infinite-volume limit. We establish the entropic repulsion between macroscopic domain walls (boundaries of level sets), showing that they are maximally separated in a precise sense described by a rotation-of-the-circle dynamical system. Lastly, we show exponential decay of correlations in the extremal components. Our setup includes, as a special case, infinitely many zero-slope integer-valued surfaces (of dimension two or higher) conditioned not to cross. We deduce the existence and structural properties of the bulk Gibbs measure over such non-crossing surfaces with any given average spacing.
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BibTeX
@misc{adhikari2026rigidity,
title = {{On the rigidity of sloped height functions in $d\ge 3$ and non-crossing surfaces}},
author = {Arka Adhikari and Reza Gheissari and Ron Peled},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.01499},
eprint = {2609.01499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01499},
}
RIS
TY - UNPB
TI - On the rigidity of sloped height functions in d≥ 3 and non-crossing surfaces
AU - Arka Adhikari
AU - Reza Gheissari
AU - Ron Peled
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.01499
UR - https://arxiv.org/abs/2609.01499
AB - We consider integer-valued ∇ϕ height functions, with general convex interactions, placed on a slope. Sheffield (2003) conjectured that for all slopes, such height functions are localized in dimensions d≥3, in the sense of having tight fluctuations in finite volume, and admitting infinite-volume limits. We establish this conjecture when there are two coordinates on which the slope vector is zero and on which the interactions are even and low temperature. In this setting, we also prove the uniqueness, in the appropriate sense, of the infinite-volume limit and classify its extremal components. We further study the structural properties of the infinite-volume limit. We establish the entropic repulsion between macroscopic domain walls (boundaries of level sets), showing that they are maximally separated in a precise sense described by a rotation-of-the-circle dynamical system. Lastly, we show exponential decay of correlations in the extremal components. Our setup includes, as a special case, infinitely many zero-slope integer-valued surfaces (of dimension two or higher) conditioned not to cross. We deduce the existence and structural properties of the bulk Gibbs measure over such non-crossing surfaces with any given average spacing.
ER -