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

Abstract

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 Zd, valid in boxes of roughly side length N−Nδ 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 Zd. This leads to a series of applications related to level-set percolation.