the incipient infinite cluster
Exactly at criticality there is (on Z^2 and conjecturally elsewhere) no infinite cluster — theta(p_c) = 0. Yet critical clusters are huge and fractal, and one would like a single canonical infinite object that captures the geometry 'at the brink of percolating'. The incipient infinite cluster (IIC) is that object: the critical cluster of the origin, made infinite by an appropriate conditioning limit, even though no infinite cluster exists unconditionally. It is the right stage on which to study transport and random walk at criticality.
Since one cannot simply condition on the zero-probability event '|C| = infinity at p_c', the IIC is built as a limit of conditional measures that each have positive probability. Kesten (1986) gave two constructions for two-dimensional percolation and proved they agree: condition critical percolation on the origin being connected to the boundary of the box of radius n, and let n tend to infinity; or condition on the origin being connected to a far-away point x and let x go to infinity. The resulting probability measure on configurations is the law of the IIC. Under it the cluster of the origin is almost surely infinite, but it is a thin, fractal object — its mass inside a box of radius R grows like R^(d_f) with a fractal dimension d_f < d (in 2D, d_f = 91/48), so it occupies vanishing volume fraction. Van der Hofstad and Jarai later constructed the IIC in high dimensions via the lace expansion, where it looks like a critical branching-random-walk trace.
The IIC matters because it is the natural geometry for studying dynamics at criticality — in particular the 'ant in the labyrinth', random walk on the critical cluster, whose subdiffusive behaviour (the Alexander-Orbach conjecture, proved in high dimensions and on trees) is one of the cleanest statements about anomalous transport in disordered media. The caveats: the IIC is genuinely a conditioning-limit construct, not an ordinary cluster, and that different reasonable conditionings give the SAME limit is a theorem, not a triviality. Its existence and uniqueness are established in 2D and in high dimensions, but for intermediate dimensions, where even theta(p_c) = 0 is not fully proved, the IIC's status inherits those gaps.
On Z^2, sample critical percolation conditioned to connect the origin to the boundary of a box of side 1000. The origin's cluster is now a sprawling, lacy backbone with dangling ends — a finite-n snapshot of the IIC. As the box grows, this picture converges to a genuinely infinite fractal cluster of dimension 91/48 on which a random walker diffuses anomalously slowly.
The IIC: a critical cluster made infinite by conditioning, on which transport is anomalously slow.
The IIC is a conditioning-LIMIT, not an ordinary cluster: at p_c the unconditional infinite cluster does not exist (theta(p_c) = 0). That distinct natural conditionings yield the same IIC is a theorem of Kesten, not a triviality, and is rigorous so far only in 2D and high dimensions.