Incipient Spanning Clusters Research Articles

Incipient infinite percolation clusters in 2D

We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for incipient spanning clusters, large clusters in a finite box and the inhomogeneous model of Chayes, Chayes and Durrett. Our results prove the equivalence of several natural definitions of the IIC. We also show that for any k≥1 the difference in size between the kth and (k+1)st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. In addition, the distribution of the Chayes--Chayes--Durrett cluster is shown to be singular with respect to the IIC.

Probability of incipient spanning clusters in critical two-dimensional percolation

The probability of simultaneous occurence of at least κ spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice at the bond percolation threshold p c = 1 2 . The calculated probabilities for free boundary conditions and periodic boundary conditions are in a very good coincidence with the exact formulae developed recently by Cardy.

The number of incipient spanning clusters in two-dimensional percolation

Using methods of conformal field theory, we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k, in the limit when k is large.

Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

The probability of simultaneous occurrence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice with free boundaries at the bond percolation threshold pc =1/2. It is found that the probability of k and more Incipient Spanning Clusters (ISC) have the values P(k>1) ≈ 0.00658(3) and P(k>2) ≈ 0.00000148(21) provided that the limit of these probabilities for infinite lattices exists. The probability P(k>3) of more than three ISC could be estimated to be of the order of 10-11 and is beyond the possibility to compute such a value by nowadays computers. So, it is impossible to check in simulations the Aizenman law for the probabilities when k≫1. We have detected a single sample with four ISC in a total number of about 1010 samples investigated. The probability of this single event is 1/10 for that number of samples. The influence of boundary conditions is discussed in the last section.

On the number of incipient spanning clusters

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 − αk 2 . In dimensions d > 6, when η = 0 the spanning clusters proliferate: for L → ∞ the spanning probability tends to one, and there typically are ≈ L d−6 spanning clusters of size comparable to | C max| 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.