The Incipient Infinite Cluster for High-Dimensional Unoriented Percolation

We consider bond percolation on $\mathbb{Z}^d$ at the critical occupation density p _c for d>6 in two different models. The first is the nearest-neighbor model in dimension d?6. The second model is a “spread-out” model having long range parameterized by L in dimension d>6. In the spread-out case, we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as |x|→∞ when d>6 and L is sufficiently large. We also give a general criterion for this convergence to hold, which is satisfied in the case d?6 in the nearest-neighbor model by work of Hara.⁽¹²⁾ We further give a second construction, by taking p<p _c , defining a measure $\mathbb{Q}^p $ and taking its limit as p↗p ^? _c . The limiting object is the high-dimensional analogue of Kesten's incipient infinite cluster (IIC) in d=2. We also investigate properties of the IIC such as bounds on the growth rate of the cluster that show its four-dimensional nature. The proofs of both the existence and of the claimed properties of the IIC use the lace expansion. Finally, we give heuristics connecting the incipient infinite cluster to invasion percolation, and use this connection to support the well-known conjecture that for d>6 the probability for invasion percolation to reach a site x is asymptotic to c|x|^?(d?4) as |x|→∞.