Phase transitions in the hard-wall large deviations for the Gaussian free field

We study the probability that the Gaussian free field on Zᵈ, d≥3, stays above a level -hN throughout the lattice N-blow-up of a bounded Lipschitz domain, when hN is close to the entropic-repulsion height √4g₀log N. Two mechanisms compete: a collective upward displacement of the field, whose cost is governed by capacity, and the suppression of the remaining local violations, whose cost is their expected number, both of which are of order Nᵈ⁻²/log N exactly in the window hN(β)²=g₀(4log N+loglog N+2β). We obtain a sharp asymptotic rate expressed through a variational problem combining a Dirichlet energy with an exponential potential inside this critical window. Outside this window we obtain the asymptotics at the appropriate moving speeds: capacity alone governs the cost as β→-∞, and the expected number of violations as β→+∞. As an application, we derive lower-deviation bounds for the cover level of a macroscopic body by random interlacements, which determine the order of decay at the transition.