Sharp connectivity bounds for the vacant set of random interlacements

We consider percolation of the vacant set of random interlacements at intensity u in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of u>0. These bounds are sharp up to principal exponential order for all u in dimension three and all u≠u∗ in higher dimensions, where u∗ refers to the critical parameter of the model, and they match the upper bounds derived in the article (Goswami, Rodriguez, Shulzhenko (2025)). In dimension three, our results further imply that the truncated two-point function grows at large distances x at a rate that depends on x only through its Euclidean norm, which offers a glimpse of the expected (Euclidean) invariance of the scaling limit at criticality. The decay rate is atypical, it incurs a logarithmic correction and comes with an explicit pre-factor that converges to 0 as the parameter u approaches the critical point u∗ from either side. A particular challenge stems from the combined effects of lack of monotonicity due to the truncation in the super-critical phase, and the precise (rotationally invariant) controls we seek, that measure the effects of a certain “harmonic humpback” function. Among others, their derivation relies on rather fine estimates for hitting probabilities of the random walk in arbitrary direction e, which witness this invariance at the discrete level, and preclude straightforward applications of projection arguments.