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Arctic boundaries of the ice model on three-bundle domains


In this paper we consider the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo and Sportiello (J Stat Phys 164:1488–1523, 2016) as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.

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  1. 1.

    Here, we always orient the edges of \({\mathbb {Z}}^2\) to the north or to the east; this makes \({\mathbb {Z}}^2\) into a directed graph.

  2. 2.

    This discrepancy arises from the fact that slightly different symmetries are required to transform the southeast part of the arctic boundary to its other parts in the cases of the three-bundle domain \({\mathcal {T}}\) and the square domain \({\mathcal {S}}\).

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    One might view \(-\infty \) and \(\infty \) as consisting of the unique vertices \(\big \{ (-\infty , -\infty ) \big \}\) and \(\big \{ (\infty , \infty ) \big \}\), respectively.


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The author heartily thanks Filippo Colomo and Andrea Sportiello for stimulating conversations and enlightening explanations, as well as Alexei Borodin for valuable encouragement and helpful discussions and suggestions. The author would also like to thank two anonymous referees for their helpful comments on an earlier draft of this paper. The author is additionally grateful to the workshop, “Conference on Quantum Integrable Systems, Conformal Field Theories and Stochastic Processes,” held in 2016 at the Institut d’Études Scientifiques de Cargése (funded by NSF Grant DMS:1637087), where he was introduced to the tangent method. This work was partially supported by the NSF Graduate Research Fellowship under Grant Number DGE1144152.

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Aggarwal, A. Arctic boundaries of the ice model on three-bundle domains. Invent. math. 220, 611–671 (2020).

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