Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

TL;DR

This study uses Monte Carlo simulations to estimate the probabilities of multiple incipient spanning clusters at the critical bond percolation threshold on a 2D square lattice, revealing extremely low probabilities for higher numbers of clusters.

Contribution

It provides the first detailed estimates of the probabilities for multiple incipient spanning clusters at criticality, highlighting the rarity of such events and limitations of current computational methods.

Findings

Probability of >1 ISC is approximately 0.00658.

Probability of >2 ISC is approximately 1.48e-6.

Event of >3 ISC is estimated around 1e-11, beyond current computational verification.

Abstract

The probability of simultaneous occurence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice at the bond percolation threshold $p_c=1/2$. It is found that the probability of k and more Incipient Spanning Clusters (ISC) has the values $P(k>1) \approx 0.00658(3)$ and $P(k>2) \approx 0.00000148(21)$ 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 a such value by nowdays 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 4 ISC in a total number of about 10^{10} samples investigated. The probability of single event is 1/10 for that number of samples.