On the number of incipient spanning clusters

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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.

Original languageEnglish (US)
Pages (from-to)551-582
Number of pages32
JournalNuclear Physics B
Volume485
Issue number3
DOIs
StatePublished - Feb 10 1997

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics

Keywords

  • Critical behavior
  • Hyperscaling
  • Incipient infinite cluster
  • Incipient spanning clusters
  • Percolation
  • Scaling limit

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