The effect of free boundary conditions on the Ising model in high dimensions

Federico Camia; Jianping Jiang; Charles M. Newman

doi:10.1007/s00440-021-01041-9

The effect of free boundary conditions on the Ising model in high dimensions

Federico Camia, Jianping Jiang^*, Charles M. Newman

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

We study the critical Ising model with free boundary conditions on finite domains in Z^d with d≥ 4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order | x- y| ^-⁽^d^-²⁾ for large | x- y| , we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L² as L→ ∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.

Original language	English
Pages (from-to)	311-328
Number of pages	18
Journal	Probability Theory and Related Fields
Volume	181
Issue number	1-3
Early online date	31 Mar 2021
DOIs	https://doi.org/10.1007/s00440-021-01041-9
Publication status	Published - Nov 2021

Bibliographical note

Funding Information:
The research of the second author was partially supported by NSFC grant 11901394 and that of the third author by US-NSF grant DMS-1507019. The authors thank Akira Sakai and Gordon Slade for useful comments. The authors also thank an anonymous referee for a very careful reading of the paper.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

Funding

The research of the second author was partially supported by NSFC grant 11901394 and that of the third author by US-NSF grant DMS-1507019. The authors thank Akira Sakai and Gordon Slade for useful comments. The authors also thank an anonymous referee for a very careful reading of the paper.

Funders	Funder number
National Science Foundation	1507019
US-NSF	DMS-1507019
National Natural Science Foundation of China	11901394

Keywords

Correlation decay
Free boundary conditions
High dimensions
Ising model
Near-critical
Susceptibility

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title = "The effect of free boundary conditions on the Ising model in high dimensions",

abstract = "We study the critical Ising model with free boundary conditions on finite domains in Zd with d≥ 4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order | x- y| -(d-2) for large | x- y| , we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L2 as L→ ∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.",

keywords = "Correlation decay, Free boundary conditions, High dimensions, Ising model, Near-critical, Susceptibility",

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PY - 2021/11

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N2 - We study the critical Ising model with free boundary conditions on finite domains in Zd with d≥ 4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order | x- y| -(d-2) for large | x- y| , we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L2 as L→ ∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.

AB - We study the critical Ising model with free boundary conditions on finite domains in Zd with d≥ 4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order | x- y| -(d-2) for large | x- y| , we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L2 as L→ ∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.

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KW - Near-critical

KW - Susceptibility

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The effect of free boundary conditions on the Ising model in high dimensions

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Keywords

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