2D Site Percolation

What you are seeing: a square lattice where each site is independently "occupied" with probability $p$ . As $p$ rises past a critical value, a single giant connected cluster appears that spans the lattice from top to bottom (the percolating cluster). Below the critical $p$ , only small isolated islands.

Critical probability $p_c = 0.59274621\ldots$ . Drag the slider through $p_c$ and watch the landscape change from "small puddles" to "one continent". The largest-cluster fraction is shown live; if a spanning cluster exists its sites are highlighted in yellow.

Figure 1. 2D site percolation with union-find cluster labeling .

p0.590

L80

WHAT TO TRY

Sweep the occupation p across p_c = 0.5927: below it the lattice is a confetti of small clusters and the order parameter sits near zero; cross it and one gold spanning cluster suddenly links top to bottom.
Park p exactly at p_c (the button): the cluster-size distribution stretches into a scale-free power law, clusters of every size, the fingerprint of a continuous phase transition.
Resample at fixed p near threshold: whether a spanning cluster forms flickers between yes and no, the finite-size critical fluctuations that sharpen into a step only as the lattice grows.