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

Research output: Contribution to JournalArticleAcademicpeer-review


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.

Original languageEnglish
Pages (from-to)311-328
Number of pages18
JournalProbability Theory and Related Fields
Issue number1-3
Early online date31 Mar 2021
Publication statusPublished - 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.


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


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