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Logarithmic Variance for the Height Function of Square-Ice

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Abstract

In this article, we prove that the height function associated with the square-ice model (i.e. the six-vertex model with \(a=b=c=1\) on the square lattice), or, equivalently, of the uniform random homomorphisms from \(\mathbb {Z}^2\) to \(\mathbb {Z}\), has logarithmic variance. This establishes a strong form of roughness of this height function.

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Notes

  1. Technically, such a distribution is only defined up to a translation — we will assume \(h_u = 0\) for some fixed u, and note that all terms in the theorem below do not depend on the choice of u.

  2. The continuous path is made by joining the vertices by straight lines in \({\mathbb {R}}^2\).

  3. We prefer the use of the square-root trick to the use of the union bound since we will refer to this argument later with events having a probability close to 1. We recall that the square-root trick yields that for increasing events \(\mathcal {A}_,\dots ,\mathcal {A}_s\) and a measure \({\mathbb {P}}\) satisfying the FKG inequality,

    $$\begin{aligned} \max _{i\le s}{\mathbb {P}}[\mathcal {A}_i]\ge 1-(1-{\mathbb {P}}[\mathcal {A}_1\cup \dots \cup \mathcal {A}_s])^{1/s}. \end{aligned}$$

  4. In (3.10), we needed to apply Proposition 3.1 twice and Proposition 3.2 once which gave a factor of \(\frac{1}{32}\) instead of \(\frac{1}{8}\).

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Acknowledgements

The first author is supported by the ERC CriBLaM, the NCCR SwissMAP, the Swiss NSF and an IDEX Chair from Paris-Saclay. The second author was supported in part by the European Research Council starting Grant 678520 (LocalOrder), and the Zuckerman STEM leadership Postdoctoral Fellowship. The last author is supported in part by NSERC 50311-57400. This project was initiated during the visit of the last author in IHES. The authors would like to express their gratitude to IHES for its support. Finally, we thank Alex Karrila and the anonymous referee for carefully reading the manuscript. After the preparation of this article, question 1.5 has been solved in [10] exploiting a key input coming from Bethe Ansatz calculations.

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Authors and Affiliations

  1. IHES, Université de Genève, Geneva, Switzerland

    Hugo Duminil-Copin

  2. University of Tel Aviv, Tel Aviv, Israel

    Matan Harel

  3. Université de Paris–Diderot, Paris, France

    Benoit Laslier

  4. ETH, Zürich, Switzerland

    Aran Raoufi

  5. University of Victoria, Victoria, Canada

    Gourab Ray

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  1. Hugo Duminil-Copin
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Correspondence to Gourab Ray.

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Communicated by J. Ding.

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Duminil-Copin, H., Harel, M., Laslier, B. et al. Logarithmic Variance for the Height Function of Square-Ice. Commun. Math. Phys. 396, 867–902 (2022). https://doi.org/10.1007/s00220-022-04483-x

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