This paper studies critical Bernoulli bond percolation on the integer lattice Z^d when the dimension d is large. The authors prove that if you take k distinct points in continuous space, scale them out to distance n and ask for the probability that the corresponding lattice sites all belong to the same open cluster, then that probability, after multiplying by a specific power of n, converges as n → ∞ to an explicit constant. This confirms a conjecture of Aizenman and Newman about the large-scale behaviour of connections in high-dimensional critical percolation.

Concretely, the limit is not a single number but a formula built from a few ingredients. One ingredient, called α in the paper, summarizes how the two-point connectivity probability decays at long distances. Another ingredient, called ρ, is a “vertex factor” that comes from a non-intersection probability for three independent samples of the incipient infinite cluster (IIC) — a probabilistic object that captures the geometry of a critical cluster seen at large scales. The final formula is a finite sum over binary branching trees, each term involving an explicit multidimensional integral that depends on the locations of the k points.

At a high level the proof proceeds by induction on k, with the three-point case as the base. The authors first show that, with high probability, the minimal way the k points can be connected looks like a binary branching tree. They then use a switching trick that removes two leaves of this tree by locating a last pivotal edge (an edge that every connecting path must use) and replacing an event where that edge is open by a related event where it is closed. After this switch one can decouple the configuration into parts that are handled by the IIC measure. Repeated application of these steps reduces the k-point problem to lower-order cases and produces the factors α and ρ. The sum over possible positions of the pivotal points then becomes a Riemann sum that converges to the integrals in the final formula.