This is the same neutral-landscape percolation model as the 3D grid view, but instead of only rendering one landscape at a time, the top panel keeps generating small independent 33×33 landscapes in the background — at the habitat fraction of every 5%-wide bucket along the x-axis, using your current spatial-clustering (Hurst) value — and plots what fraction of them percolate. That turns a single realization into a genuine phase-transition curve: a statistical object the 3D view cannot show at all, since it only ever displays one fractal mosaic per frame.
1. Elevation field via midpoint displacement (diamond–square),
roughness set by Hurst exponent H: scale(step) = scale(step/2)·2^(−H)
2. Threshold the field at the percentile matching fraction p
→ binary habitat/matrix grid (identical to the bottom panel's map)
3. Flood-fill (BFS, 4-neighbour) labels every connected cluster;
it "percolates" if the largest one touches two opposite edges.
4. Repeat step 1-3 many times per p-bucket → P(percolate | p, H)
accumulates into the curve — its 50%-crossing IS the empirical
percolation threshold p꜀ for the current H.
For an uncorrelated random landscape (H → 0) on a square lattice, classic percolation theory predicts a sharp transition at the critical site-occupation probability p꜀ ≈ 59.3% (dashed reference line) — below it the curve sits near 0, above it near 1, with a narrow crossover band that sharpens as the sampling grid grows. Raising H shifts the whole curve left: because midpoint displacement clumps habitat into fewer, larger contiguous patches for the same p, a correlated landscape percolates at a lower habitat fraction than a random one — the same real-world fact that lets species keep moving across landscapes with correlated terrain (elevation, soil, moisture) even after more clearing than a random-loss model would predict.
- Habitat fraction p — moves the vertical marker on the curve and rebuilds the mosaic map below at that fraction.
- Spatial clustering H — resets the Monte-Carlo curve (it depends on H) and rebuilds the mosaic map.
- Grid resolution — resolution of the mosaic map only; the Monte-Carlo curve always samples a fixed 33×33 grid so it can accumulate hundreds of trials per second.
- New landscape — reseeds the mosaic map with a fresh random fractal field at the same p and H (does not affect the curve, which already averages over many seeds).