On the Number of Incipient Spanning Clusters

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