Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters

Abstract: We consider the level sets of the Gaussian free field and the vacant set of random interlacements, both defined on a typical realization of the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}<sup>d$, $d \geq 3$. We prove that in the entire supercritical regime of Bernoulli bond percolation, both the level sets of the Gaussian free field and the vacant set of random interlacements undergo non-trivial percolation phase transitions at deterministic critical levels. A key aspect of the proof is the development of certain quenched controls over tree embeddings, permitting the application of a static renormalization scheme in the presence of spatial irregularities, which may be of independent interest.