Limit shapes and harmonic tricks

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.