On the number of incipient spanning clusters

Abstract

In critical percolation models, in a large cube there will typically be more than one cluster of comparable diameter. In 2D, the probability of k ⪢ 1 spanning clusters is of the order e−αk2. In dimensions d > 6, when η = 0 the spanning clusters proliferate: for L → ∞ the spanning probability tends to one, and there typically are ≈ Ld−6 spanning clusters of size comparable to |Cmax|L4. The rigorous results confirm a generally accepted picture for d > 6, but also correct some misconceptions concerning the uniqueness of the dominant cluster. We distinguish between two related concepts: the Incipient Infinite Cluster, which is unique partly due to its construction, and the Incipient Spanning Clusters, which are not. The scaling limits of the ISC show interesting differences between low (d = 2) and high dimensions. In the latter case (d > 6?) we find indication that the double limit: infinite volume and zero lattice spacing, when properly defined would exhibit both percolation at the critical state and infinitely many infinite clusters.