On large deviations and intersection of random interlacements

We investigate random interlacements on Zd with d≥3, and derive the large deviation rate for the probability that the capacity of the interlacement set in a macroscopic box is much smaller than that of the box. As an application, we obtain the large deviation rate for the probability that two independent interlacements have empty intersections in a macroscopic box. Additionally, we prove that conditioning on this event, one of the interlacements will be sparse in terms of capacity within the box. This result is an example of the entropic repulsion phenomenon for random interlacements.