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In this article we obtain for d ≥ 3 an approximation of the zero-average Gaussian free field on the discrete d-dimensional torus of large side length N by the Gaussian free field on Z d , valid in boxes of roughly side length N − N δ with δ ∈ ( 1 2 , 1). As an implication, the level sets of the zero-average Gaussian free field on the torus can be approximated by the level sets of the Gaussian free field on Z d . This leads to a series of applications related to level-set percolation.

We study level-set percolation of the Gaussian free field on the infinite d-regular tree for fixed d ≥ 3. Denoting by h the critical value, we obtain the following results: for h > h we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level h; for h < h we prove that the number of vertices connected over distance k above level h to a fixed vertex grows exponentially in k with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level h, at least away from the critical value h . Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value h and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [A Č19].