Random walks, disconnection and random interlacements

This work is about the disconnection of graphs by trajectories of random walks. Computer simulations show that the components left in a large graph after removing the trajectory of a simple random walk of a suitable timescale exhibit interesting phase transitions, not unlike the ones encountered in the widely studied field of random graphs. Disconnection phenomena of this kind are only beginning to be understood at a mathematically rigorous level. This thesis contributes to the field in several directions. The author studies the influence of a bias on the disconnection time of a discrete cylinder by a random walk, the vacant set left by a random walk on a discrete torus, and the link between random walk trajectories performing disconnection and the model of random interlacements, whereby questions on disconnection are related to problems in percolation theory.