Correlation Functions in Percolation

Site-Site Correlation Function

The site-site correlation function g(ri,rj) measures the probability that two sites belong to the same finite cluster. For one-dimensional percolation:

$$g(r_i,r_j) = p^r \quad \text{for } 0 < p < 1$$

where r = |ri - rj| is the distance between sites in lattice units

p = 0.80

Correlation Length

The correlation length ξ(p) characterizes the decay of the correlation function:

$$g(r_i,r_j) = \exp(-r/\xi)$$$$\xi(p) = -\frac{1}{\ln p} \to (1-p)^{-1} \quad \text{for } p \to 1^-$$

Connection to Average Cluster Size

The correlation function is directly related to the average cluster size χ(p):

$$\sum_{r_j} g(r_i,r_j) = \cdots + p^2 + p^1 + p^0 + p^1 + p^2 + \cdots = \frac{1+p}{1-p} = \chi(p)$$

This geometric series sums to give exactly the average cluster size we found earlier, demonstrating the deep connection between spatial correlations and cluster properties.

Key Points:

Correlation function decays exponentially with distance
Correlation length diverges as p approaches 1
Sum of correlations equals average cluster size
These relationships hold for one-dimensional percolation