Let ($Σ$, g) be a closed connected surface equipped with a riemannian metric. Let ($λ$ n) n$\in$N and ($ψ$ n) n$\in$N be the increasing sequence of eigenvalues and the sequence of corresponding L 2-normalized eigenfunctions of the laplacian on $Σ$. For each L \textgreater{} 0, we consider $φ$ L = 0\textless{}$λ$n$\le$L $ξ$n $\sqrt$ $λ$n $ψ$ n where the $ξ$ n are i.i.d centered gaussians with variance 1. As L $\rightarrow$ $\infty$, $φ$ L converges a.s. to the Gaussian Free Field on $Σ$ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L $\rightarrow$ $\infty$. We then use this result to obtain the asymptotics of the probability that $φ$ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of $φ$ L around 1 $\sqrt$ 2$π$ ln L.