Entropic repulsion to the middle layer
Abstract
We consider ∇φ height functions with even and convex interaction energy W on the lattice ℤd, which are restricted to take values in the set {−S, …, S} for some integer S ≥ 1. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension d=2 at all temperatures. (b) Dimensions d≥3 at all temperatures, for a wide class of W with non-increasing second derivative, including the family W(x)=|x|p for p∈[1,2]. (c) Dimensions d≥ 3 at both low and high temperatures, T∈ (0,W(1)d/(4(ln d+ln 8)))∪(d W(2S),∞], with the normalization W(0)=0. At low temperatures, our proof provides an alternative to Pirogov--Sinai methods. Conversely, we exhibit a class of even and convex interaction energies W which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in (−1/2,1/2). This implies the delocalization of the model restricted to take values in {0,1,…} (i.e., conditioned to lie above a floor) for all dimensions, any even and convex W, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval [−1,1] always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in {−S+1/2, …, S−1/2}, $S \ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in (−1,1). The special case W(x)=x2 of our results addresses questions left open in the work of Bricmont--El Mellouki--Fröhlich (1986).
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BibTeX
@misc{mihailescu2026entropic,
title = {{Entropic repulsion to the middle layer}},
author = {Max Mihailescu and Ron Peled},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07578},
eprint = {2609.07578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07578},
}
RIS
TY - UNPB
TI - Entropic repulsion to the middle layer
AU - Max Mihailescu
AU - Ron Peled
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07578
UR - https://arxiv.org/abs/2609.07578
AB - We consider ∇φ height functions with even and convex interaction energy W on the lattice ℤ^d, which are restricted to take values in the set {−S, …, S} for some integer S ≥ 1. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension d=2 at all temperatures. (b) Dimensions d≥3 at all temperatures, for a wide class of W with non-increasing second derivative, including the family W(x)=|x|^p for p∈[1,2]. (c) Dimensions d≥ 3 at both low and high temperatures, T∈ (0,W(1)d/(4(ln d+ln 8)))∪(d W(2S),∞], with the normalization W(0)=0. At low temperatures, our proof provides an alternative to Pirogov--Sinai methods. Conversely, we exhibit a class of even and convex interaction energies W which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in (−1/2,1/2). This implies the delocalization of the model restricted to take values in {0,1,…} (i.e., conditioned to lie above a floor) for all dimensions, any even and convex W, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval [−1,1] always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in {−S+1/2, …, S−1/2}, $S \ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in (−1,1). The special case W(x)=x^2 of our results addresses questions left open in the work of Bricmont--El Mellouki--Fröhlich (1986).
ER -