Artificial Wasteland · research · percolation-played

The math behind Percolation, as Texture

The verification notebook for the instrument Percolation, as Texture — the connectivity phase transition sonified as a drone whose spectrum is the live cluster-size distribution. Everything the page plays is proved here, building on the threshold and self-duality checks in research/percolation.

A · the threshold markers are honest

• The spanning probability crosses ½ at each lattice's cited p_c (union-find, bisection, finite-size L=64):

site □:  spans-½ at p ≈ 0.5929   cited p_c = 0.59274605079210   no closed form (Jacobsen 2015)
bond □:  spans-½ at p ≈ 0.4999   cited p_c = ½ exactly            (Harris 1960 / Kesten 1980)
site △:  spans-½ at p ≈ 0.5005   cited p_c = ½ exactly            (self-matching)

Only two of the three are provably exact. The site-□ marker is drawn at the measured value and never claimed exact — no closed form is known, and it may have none.

B · the order parameter snaps up at p_c (the bass tone)

• P∞ = (largest cluster)/N is ≈ 0 below p_c and rises steeply above — site □: 0.01 → 0.74 across the transition; bond □: 0.02 → 0.96 across ½. The deep partial the ear hears enter is this number.

C · the spectrum is the distribution

• The page bins clusters by log₂(size) and drives one tone per bin at amplitude ∝ that bin's share of the grid. The binning conserves mass (every clustered cell counted exactly once), the giant lands in a low bin (low tone), singletons in bin 0 (high tone).

reproduce

node research/percolation-played/verify.mjs                        # 11 checks + writes embed.js
npm run build && node research/percolation-played/verify-page.mjs  # 22 checks ×2, the page

Sources: Broadbent–Hammersley, Proc. Camb. Phil. Soc. 53 (1957) · Harris (1960) / Kesten, Comm. Math. Phys. 74 (1980) · Jacobsen, J. Phys. A 48 (2015) · Stauffer–Aharony, Introduction to Percolation Theory (1994).