On the Number of Incipient Spanning Clusters

TL;DR

This paper investigates the behavior and number of spanning clusters in critical percolation models across different dimensions, revealing that high dimensions exhibit multiple incipient spanning clusters and correcting misconceptions about cluster uniqueness.

Contribution

It provides rigorous results on the proliferation of spanning clusters in high-dimensional percolation, clarifies the distinction between Incipient Infinite Clusters and Incipient Spanning Clusters, and discusses their scaling limits.

Findings

In dimensions d>6, spanning clusters proliferate with approximately L^{d-6} clusters.

In 2D, the probability of multiple spanning clusters decreases exponentially with the number of clusters.

High-dimensional models suggest the coexistence of percolation at criticality and infinitely many clusters in the double limit.

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 $e^{-\alpha k^{2}}$. In dimensions d>6, when $\eta = 0$ the spanning clusters proliferate: for $L\to \infty$ the spanning probability tends to one, and there typically are $ \approx L^{d-6}$ spanning clusters of size comparable to $|\C_{max}| \approx L^4$. 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.