The random-cluster model is defined on subgraphs of $\mathbb{Z}^2$ and has two parameters: cluster-weight $q>0$ and edge-probability $0<p<1$. It is classical that, for each $q\geq 1$, the model undergoes a percolation phase transition when $p=p_c(q)$. Beffara and Duminil-Copin in 2010 computed $p_c(q)$, and later works established the type of the phase transition: it is continuous when $1 \leq q \leq 4$ and discontinuous when $q>4$. The former is characterised by Russo-Seymour-Welsh estimates, while the latter asserts non-uniqueness of the infinite-volume DLR/Gibbs measure.

In this talk we revisit both parts of this diagram. When $1 \leq q \leq 4$, we give a new proof of continuity that does not use parafermionic observable, nor Bethe Ansatz. When $q>4$, we establish invariance principle under Dobrushin boundary conditions: the interface converges to the Brownian bridge. Both arguments rely on the Baxter-Kelland-Wu correspondence that relates the random-cluster model to a certain height function (six-vertex model). Remarkably, we obtain also some result when $q<1$, though only at the self-dual point.