The Rounding Crystal

In FK-percolation on the square lattice with cluster weight q > 4, large clusters adopt an equilibrium shape — the Wulff crystal. The lattice imposes anisotropy: the crystal has flat faces aligned with the lattice axes, angular corners, a geometry that reflects the underlying discrete structure.

Manolescu and Mohanarangan show that as q decreases toward 4, the Wulff crystal deforms continuously into a circle. At q = 4 — the critical boundary between discontinuous and continuous phase transitions — the lattice anisotropy vanishes completely. The crystal becomes round.

The through-claim: the lattice model forgets it lives on a lattice exactly at the critical point. The geometric signature of discrete structure — angular facets, preferred orientations — smooths away as the system approaches criticality. Rotational invariance, which is a property of continuous systems, emerges from a model defined on a square grid.

This is not a coarse-graining result (zooming out far enough that the lattice details blur). It is a genuine deformation of the equilibrium shape: the Wulff crystal’s boundary, computed exactly from the surface tension, becomes isotropic at q = 4. The discreteness of the lattice is a relevant perturbation above q = 4 and an irrelevant one at q = 4.

The crystal rounds because the critical point has already decided that the lattice does not matter.