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

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We investigate level-set percolation of the discrete Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the level set of the Gaussian free field below a level $\alpha$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ 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. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the Gaussian free field experiences an entropic push down proportional to the harmonic potential of the set $A$. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. (Joint work with M. Nitzschner)

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