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For $d \geq 3$ we obtain an approximation of the zero-average Gaussian freefield on the discrete $d$-dimensional torus of large side length $N$ by theGaussian free field on $\mathbb Z^d$, valid in boxes of roughly side length $N- N^\delta$ with $\delta \in (\frac12,1)$. As an implication, the level sets ofthe zero-average Gaussian free field on the torus can be approximated by thelevel sets of the Gaussian free field on $\mathbb Z^d$. This leads to a seriesof applications related to level-set percolation. We show that level sets ofthe zero-average Gaussian free field on the torus for levels $h > h_\star$(where $h_\star$ denotes the critical value for level-set percolation of theGaussian free field on $\mathbb Z^d$) with high probability contain noconnected component of volume comparable to the total volume of the torus.Moreover, level sets with $h < h_\star$ with high probability contain aconnected component of (extrinsic) diameter comparable to the torus diameter$N$. We also show that level sets of the zero-average Gaussian free field onthe torus for levels $h$ above a second critical parameter $h_{\star\star}(\geqh_\star)$, again defined via the Gaussian free field on $\mathbb Z^d$, withhigh probability only contain connected components negligible in their sizewhen compared to the size of the torus. Similar results have been obtained byA. Teixeira and D. Windisch in [Comm. Pure Appl. Math., 64(12):1599-1646, 2011]and J. \v{C}ern\'y and A. Teixeira in [Ann. Appl. Probab., 26(5):2883-2914,2016] for the vacant set of simple random walk on a large discrete torus withthe help of random interlacements on $\mathbb Z^d$, introduced by A.-S.Sznitman in [Ann. of Math. (2), 171(3):2039-2087, 2010].

Local picture and level-set percolation of the Gaussian free field on a large discrete torus