Final Report Summary - RWPERCRI (Random Walks, Percolation and Random Interlacements)
The investigation of random interlacements is a central topic of this research project. Random interlacements consist of a Poisson cloud of doubly-infinite random walk trajectories. They offer a microscopic model for the structure left at suitable time scales by random walks on large recurrent graphs, which are locally transient. In this research project, decoupling inequalities have been developed to cope with the long range dependence of the model. They have been applied to the derivation of quantitative controls on the percolative properties of the occupied set and the vacant set left by random interlacements, and to the derivation of a more detailed description of the geometric properties of random interlacements. Probabilistic couplings with random walk on a discrete cylinder with a large basis, and with random walk on a large discrete torus have been established. They have provided a transfer mechanism between random interlacements and random walks. For instance, this has been applied to study the presence, or the absence, of a giant component in the complement of the trajectory of a random walk on a large d-dimensional torus (d ≥3), at times proportional to the numbers of sites in the torus, when this proportionality factor varies. Deep links between random interlacements and the discrete Gaussian free field, and also with Poisson clouds of Markovian loops have been found. This novel insight has permitted to apply interlacements methodology to find the solution to questions dating back to the eighties concerning the percolative properties of the level sets of the Gaussian free field. An isomorphism theorem, which relates the random field of occupation-times of random interlacements to the Gaussian free field, has been established. Large deviation principles for the random field of occupation-times of random interlacements have been obtained, and when the vacant set is in the percolative regime, these controls have been applied to study the fashion in which random interlacements can disconnect large bodies from infinity.