Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

We investigate level-set percolation of the discrete Gaussian free field on Zd, d≥ 3 , in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level α disconnects the discrete blow-up of a compact set A⊆ Rd from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of A, when disconnection occurs. These bounds, combined with the findings of the recent work by Duminil-Copin, Goswami, Rodriguez and Severo, show that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of A. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain ‘profile’ description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of A, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the ‘solidification estimates’ developed in Nitzschner and Sznitman (to appear in J Eur Math Soc, arXiv:1706.07229).