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Let ($\Sigma$, g) be a closed connected surface equipped with a riemannianmetric. Let ($\lambda$ n) n$\in$N and ($\psi$ n) n$\in$N be the increasingsequence of eigenvalues and the sequence of corresponding L 2-normalizedeigenfunctions of the laplacian on $\Sigma$. For each L \textgreater{} 0, weconsider $\phi$ L = 0\textless{}$\lambda$n$\le$L $\xi$n $\sqrt$ $\lambda$n$\psi$ n where the $\xi$ n are i.i.d centered gaussians with variance 1. As L$\rightarrow$ $\infty$, $\phi$ L converges a.s. to the Gaussian Free Field on$\Sigma$ in the sense of distributions. We first compute the asymptoticbehavior of the covariance function for this family of fields as L$\rightarrow$ $\infty$. We then use this result to obtain the asymptotics ofthe probability that $\phi$ L is positive on a given open proper subset withsmooth boundary. In doing so, we also prove the concentration of the supremumof $\phi$ L around 1 $\sqrt$ 2$\pi$ ln L.

Hole probability for nodal sets of the cut-off Gaussian Free Field