We consider minimizers of nonlocal functionals, like the fractional perimeter, or the fractional anisotropic perimeter, in low dimensions. It is known that a minimizer for the nonlocal perimeter in $\mathbb{R}^2 $ is necessarily an halfplane. We give a quantitative version of this result, in the following sense: we prove that minimizers in a ball of radius $R$ are nearly flat in $B_1$, when $R$ is large enough. More precisely, we establish a quantitative estimate on how "close" these sets are (in the $L^{1}$ -sense and in the $L^{\infty}$ -sense) to be a halfplane, depending on $R$. This is a joint work with Joaquim Serra and Enrico Valdinoci.