In the dimer model — perfect matchings of a planar graph — large random tilings exhibit a sharp boundary between frozen regions (where the tiling is deterministic) and liquid regions (where it fluctuates). This boundary is the arctic curve, and its shape encodes the macroscopic statistics of the random tiling.
The paper on limit shapes and harmonic tricks (arXiv: 2603.21255) extends the tangent plane method for computing arctic curves to multiply connected domains — regions with holes.
Simply connected domains (like the Aztec diamond) have arctic curves computable by complex analysis. Multiply connected domains require elliptic functions: the hole introduces a topological parameter (the height change around it) that modulates the arctic curve continuously. The frozen boundary depends on a discrete parameter — how much the tiling “wraps” around the hole — and the parametrization uses elliptic functions because the underlying Riemann surface has genus one.
This is the first explicit parametrization of arctic curves for multiply connected regions. The hole transforms the complex analysis from rational (genus zero) to elliptic (genus one), and the arctic curve becomes a family indexed by the hole height.
The through-claim: topology changes the algebra of randomness. A hole in the domain promotes the arctic curve from rational to elliptic — the genus of the boundary curve matches the genus of the domain. The macroscopic statistics of a random tiling know the topology of the space, and they know it through the algebra of the parametrizing functions.
2603.21255. Probability / dimer model / arctic curves / elliptic functions / limit shapes.