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

TL;DR

This paper investigates how the shape of the Wulff crystal in self-dual FK-percolation on the square lattice becomes round as the model approaches the critical point from the discontinuous regime, revealing isotropic correlation length behavior.

Contribution

It proves that the Wulff crystal becomes round and the correlation length becomes isotropic as q approaches 4 from above, extending recent rotational invariance results.

Findings

Wulff crystal becomes round near criticality

Correlation length becomes isotropic approaching q=4

Behavior extends previous rotational invariance results

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).