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A formula for boundary correlations of the critical Ising model

Abstract

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical Z-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula describing spin correlations between boundary vertices in terms of the shape of the region. When the region is a regular polygon, our formula becomes an explicit trigonometric sum.

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Notes

  1. By definition, we have \(e_{a}(x_1,x_2,\dots ,x_b):=\sum _{1\le j_1<j_2<\dots <j_{a}\le b} x_{j_1}x_{j_2}\cdots x_{j_{a}}\).

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Acknowledgements

I am indebted to Pasha Pylyavskyy for his numerous contributions at various stages of this project. I also thank Clément Hongler for bringing several useful references to my attention. In addition, I am grateful to the anonymous referee for their valuable suggestions. This work was supported by an Alfred P. Sloan Research Fellowship and by the National Science Foundation under Grants Nos. DMS-1954121 and DMS-2046915.

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Correspondence to Pavel Galashin.

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Galashin, P. A formula for boundary correlations of the critical Ising model. Probab. Theory Relat. Fields 182, 615–640 (2022). https://doi.org/10.1007/s00440-021-01086-w

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  • DOI: https://doi.org/10.1007/s00440-021-01086-w

Keywords

  • Critical Z-invariant Ising model
  • Totally nonnegative Grassmannian
  • Fourier transform
  • Regular polygons
  • Boundary spin correlations

Mathematics Subject Classification

  • Primary: 82B27
  • Secondary: 14M15
  • 15B48