Percolation Theory

Percolation theory studies how connectivity emerges in random systems. It has applications in epidemic spreading, oil recovery, and network resilience.

Site Percolation on a 1D Chain

In site percolation, each site on a lattice is independently occupied with probability $p$. A cluster is a maximal connected group of occupied sites.

In 1D, sites are connected only to their immediate left and right neighbors.

Percolation Threshold

The percolation threshold $p_c$ is the minimum $p$ at which an infinite cluster exists:

• 1D chain: $p_c = 1$ — you must occupy all sites for the cluster to span the system
• 2D square lattice: $p_c \approx 0.5927$
• 3D cubic lattice: $p_c \approx 0.3116$

Key Quantities

Mean cluster size (average over all clusters): $\langle s \rangle = \frac{\sum_i s_i}{\text{number of clusters}}$

Percolation probability (fraction of sites in the largest cluster): $P_{\infty} = \frac{s_{\max}}{N}$

Near the threshold: $P_{\infty} \sim (p - p_c)^\beta$ with $\beta = 1$ in 1D.

Random Number Generator

We use a Linear Congruential Generator (LCG): $x_{n+1} = (a \cdot x_n + c) \mod m$ with $a = 1664525$, $c = 1013904223$, $m = 2^{32}$. A site is occupied if $x_{n+1}/m < p$.

Implementation

def generate_1d_lattice(N, p, seed=42):
    # LCG: a=1664525, c=1013904223, m=2**32
    # Each step: state=(a*state+c)%m; occupied if state/m < p
    ...

def find_clusters_1d(lattice):
    # Scan left to right, track contiguous runs of 1s
    # Return list of cluster sizes
    ...

def mean_cluster_size(cluster_sizes):
    # Return average cluster size
    ...

def percolation_probability(cluster_sizes, N):
    # Return max(cluster_sizes) / N
    ...