What you are seeing. Training data points arrive one by one into a whitened embedding space. Each point is shown as a translucent ε-ball — the region it "covers." Two points are connected when their ε-balls overlap (Mahalanobis distance < 2ε). Connected components are found by union-find and coloured by size: blue → isolated · green → growing cluster · orange → large cluster · red → giant component.
In the K=2 phase (N below the BBP threshold slider), all points land on the blue hyperplane — the 2D eigenspace spanned by the two resolved eigenmodes. The ε-balls are translucent spheres whose colour encodes cluster size — blue for isolated, shifting through green and yellow toward red as the giant component grows. You can watch the coalescence happen: small clusters merge, the colour warms, and eventually one dominant red blob spans the plane. That is percolation in the 2D eigenspace.
At the BBP threshold (drag the N₃ slider to control when the third eigenmode opens),
the points suddenly lift off the plane into a 3D cloud.
The previously red giant component instantly becomes a thin slice of a larger 3D space —
coverage fraction drops, the colour cools back toward blue/green,
and you are back in the fragmented regime.
This is the discontinuity in γ(N) described in the theory.
The re-percolation in 3D requires substantially more points than the original
2D percolation did — because the volume to fill scales as εd,
and d just jumped from 2 to 3.
You can drag the elevation slider or click and drag the canvas to see the
3D geometry from any angle.
The core theorem.
The percolation threshold satisfies
Nc ≈ ρc(d) · Vol(Mw) / Vd(ε)
and is invariant to the eigenvalue spectrum of the raw data covariance —
it depends only on the Mahalanobis geometry and the regularisation parameter.
The BBP transition at NBBP(k) = D / (λk − 1)²
governs when the k-th eigenmode becomes statistically resolvable;
the percolation threshold then determines when that subspace is geometrically covered.