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

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Abstract

We study the critical Ising model with free boundary conditions on finite domains in \({\mathbb {Z}}^d\) with \(d\ge 4\). 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 \(L^2\) as \(L\rightarrow \infty \). 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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Acknowledgements

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.

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Authors and Affiliations

  1. Division of Science, NYU Abu Dhabi, Saadiyat Island, Abu Dhabi, UAE

    Federico Camia

  2. Department of Mathematics, Faculty of Science, Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081, HV Amsterdam, The Netherlands

    Federico Camia

  3. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Building 11, Yanqi Island, Yanqi Lake West Road, Beijing, 101408, China

    Jianping Jiang

  4. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Charles M. Newman

  5. NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China

    Charles M. Newman

Authors
  1. Federico Camia
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  2. Jianping Jiang
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  3. Charles M. Newman
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Correspondence to Jianping Jiang.

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In fond memory of Harry Kesten

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Camia, F., Jiang, J. & Newman, C.M. The effect of free boundary conditions on the Ising model in high dimensions. Probab. Theory Relat. Fields 181, 311–328 (2021). https://doi.org/10.1007/s00440-021-01041-9

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