Abstract: We study the Gibbs measures for the Potts and FK-percolation models on the square lattice. In both cases, the set of extremal Gibbs measures is known away from criticality, as well as at criticality when 
is between 1 and 4.
Our work concerns the critical case for above 4. For the Potts model, we prove that all Gibbs measures are linear combinations of the 
thermodynamic limits with free and monochromatic boundary conditions, respectively. For FK-percolation, all Gibbs measures are linear combinations of the free and wired infinite-volume measures.
The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman and Higuchi, which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation pose serious additional difficulties compared to the case of the Ising model.
Joint work with Alexander Glazman.