When stuff is about to change states, jagged "islands" of matter suddenly smooth out and become perfectly round.
March 18, 2026
Original Paper
The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality
arXiv · 2603.16318
The Takeaway
We usually think of crystals as having fixed, sharp structures until they melt or change phase. This paper proves that at a specific critical state, the directional preferences that create sharp edges vanish, forcing the entire structure to smooth out into a perfectly symmetrical sphere.
From the abstract
The study of the phase transition in planar FK-percolation on the square lattice has seen significant recent breakthroughs. The model undergoes a change in the nature of its phase transition at $q = 4$, transitioning from a continuous to a discontinuous regime. The aim of this article is to investigate the behaviour of the model in the discontinuous regime as $q > 4$ approaches the continuous transition point $4$ from above, while maintaining the critical parameter $p = p_c(q)$. We prove that in