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arXiv · 2603.21255

Limit shapes and harmonic tricks

Nikolai Kuchumov
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Abstract

This article has two main goals. First, it provides a self-contained exposition of the tangent plane method for the dimer model - a technique for analyzing arctic curves and limit shapes introduced by R. Kenyon and I. Prause (2020). Second, it extends this method to multiply connected domains through a nontrivial computation of the frozen boundary for the Aztec diamond with a hole. This computation yields the first explicit parametrization in terms of elliptic functions of a family of arctic curves of a multiply-connected region indexed by the height change (hole height). We also derive and visualize the corresponding limit height functions.

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Nikolai Kuchumov. 2026-03-22. Limit shapes and harmonic tricks. https://arxiv.org/abs/2603.21255

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