The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality

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 this limit, the correlation length becomes isotropic. The core of the proof builds upon the recently established rotational invariance of the large-scale features of the model at $q = 4$ (arXiv:2012.11672).