Surface order large deviations for 2d FK-percolation and Potts models

By adapting the renormalization techniques of Pisztora, we establish surface order large deviations estimates for FK-percolation on ${\mathbb{Z}}^2$ with parameter $q\ge 1$ and for the corresponding Potts models. Our results are valid up to the exponential decay threshold of dual connectivities which is widely believed to agree with the critical point.