Robust construction of the high-dimensional incipient infinite cluster

In Bernoulli percolation, the incipient infinite cluster (IIC) is a version of the “open cluster of the origin at criticality conditioned to be infinite”. Since this event should have probability 0 on Zd, the IIC is constructed via a limiting procedure. For d > 6, several constructions have been given and shown to produce the same object, but many natural limiting procedures remain unexplored. For instance, it is an open question whether conditioning on {0 is connected to the boundary of [-n, n]d} produces the IIC as n tends to infinity. We answer this question in the affirmative as a corollary of our theorem, which roughly says “conditioning on any long open connection produces the IIC”, and whose proof does not directly use lace expansion analysis.