Iterative Processes — Hex Path Tiles

~0.57 fraction of tiles on a loop (limit)

α ≈ 2.09 loop-size power law — barely above marginal

e^L/19 steps to reach ring L (erasing process)

s^−1.97 scale-free avalanche distribution

0.9997 density reached by random-order growth

Q7 — loop occupancy

What fraction of tiles ends up on a loop?

A finite constant — but a surprisingly subtle one. The fraction of tiles whose arc belongs to a closed loop is Σ s · ρ(s), summing loop-size × density over all sizes. From Q2 and Q5 we know ρ(s) ~ s^−α with α ≈ 2.09. That exponent is just barely above 2 — the marginal value where the sum barely converges.

Because α is only marginally above 2, the sum Σ s · ρ(s) is dominated by the rare giant loops whose contribution keeps growing with system size N. The fraction rises 0.34 at N = 50 000 → 0.41 at N = 10⁶ and is still climbing. Best estimate: f_∞ ≈ 0.55–0.60 ± 0.07, approached glacially.

Sizes 3 and 4 alone account for roughly half the loop density (ρ(3) = ρ(4) = 1/108 exactly from Q10). The rest accumulates slowly from the heavy tail — one giant loop per patch contributes as much as dozens of small ones.

Two panels: loop-tile fraction vs 1/√N showing slow upward convergence; cumulative Σ s·ρ(s) vs loop size s showing the heavy tail — **Left:** fraction of tiles on a loop vs 1/√N for spiral patches up to N ≈ 10⁶. The dashed asymptote at f ≈ 0.57 is still not reached. **Right:** cumulative Σ s·ρ(s) — the contribution from loops larger than s. The tail is heavy enough (α ≈ 2.09 barely converges) that the sum keeps growing with patch size. The size-5 gap is clearly visible.

Q11 — strand-end random walk

Why is α ≈ 2 marginal? The boundary-end picture

An open strand has exactly two ends, each sitting on a boundary edge of the growing patch. As each new ring is added, these ends can move — they random-walk on the perimeter as new tiles connect to them.

The key finding from Q11 is that this is Brownian motion in log-time τ = ln K (ring index K): the variance in an end's angular position grows linearly in τ, i.e. proportional to 1/K per ring. The single-ring step is heavy-tailed (kurtosis ~77!), but averaged over many steps the drift is diffusive in logarithmic time. Neighbouring ends on the perimeter co-move (correlation +0.55), but a strand's own two ends are nearly independent (correlation +0.06) — so the gap between them is a clean, slow random walk.

The per-layer closing law: P(close next ring | gap Δ) ≈ exp(−Δ²/2σ²) with σ ≈ 2 edges, depending only on the edge-gap and independent of perimeter length. The aggregate closing rate ≈ 0.04/strand/ring is flat as the perimeter quadruples. Strands stay short-ranged on the perimeter — they never spread around the circle. Scale-free, perimeter-independent closing → the loop-size exponent α is pinned at the marginal value 2. This also pins f_∞ ≈ 0.55 ± 0.07.

Four-panel figure: (A) end-position variance linear in log K; (B) gap distribution power law; (C) closing law exp(−Δ²/2σ²); (D) α comparison from end-meeting vs direct loop measurement — Four-panel end-walk picture. **(A)** End-position variance grows linearly in log K — plain Brownian motion in log-time. **(B)** Gap distribution between the two ends of a single strand: a power law (strands stay short-ranged, not spread around the circle). **(C)** Per-layer closing probability vs edge-gap Δ — clean Gaussian exp(−Δ²/2σ²), σ ≈ 2, independent of ring K. **(D)** The exponent reconstructed from end-meeting events matches the directly measured α ≈ 2.09 from the loop-size spectrum — confirming the mechanism.

The interactive widget below lets you watch this in real time. Pick a strand (amber = end A, pink = end B), watch the two dots wander the boundary ring by ring, and observe how the gap Δ stays small even as the perimeter grows.

Q8 / Q12 — loop-erasing growth (corrected)

A process that erases its own loops — and decelerates forever

A natural variant: place tiles in spiral order, random rotation, but whenever a placement closes a loop, delete every tile the loop passes through. Those cells go back on the free list — the earliest freed cell is always filled next.

The original analysis claimed the occupied count saturates to a steady state (compact eroded disk, late-time slope ≈ 0). Tracking the frontier — the maximum ring ever reached — reveals a different picture.

Correction
The "steady-state saturation" claim is wrong. The occupied count looks flat only because reaching each new ring takes exponentially more steps. The frontier never plateaus: it keeps advancing monotonically. Steps to reach ring L grows ≈ e^L/19 (doubling every ~13 rings). True saturation is provably impossible — a finite prefix is filled loop-free with positive probability, so the frontier passes every ring almost surely.

The process decelerates, it does not saturate. The correct reading: occupied count ∝ √steps early on, then the growth rate falls because each additional ring takes exponentially longer to reach. Both the figure below and the live widget show this directly.

Avalanches: inherited criticality

The deletion events (avalanches) have a scale-free size distribution: P(s) ∝ s^−1.97, with mean ≈ 17 and occasional giants reaching ~5000 tiles. This exponent is inherited from the loop-gas's own α ≈ 2.1 — not self-tuned. The process drives itself to a state where large, catastrophic avalanches occur at every scale, but the exponent is set by the underlying loop statistics rather than by any self-organized tuning mechanism.

A single placement can close up to 3 loops simultaneously (≈ 1% close 2, ~3 × 10⁻⁴ close 3). The avalanche size distribution remains scale-free across all these cases.

Two panels showing the loop-erasing process: frontier max-ring vs steps (monotone, still climbing) and avalanche-size PDF (power law s^{-1.97}) — **Corrected interpretation.** **Left:** max ring reached vs steps — monotone and still climbing, not saturating. The frontier creeps at an exponentially decelerating rate: steps to reach ring L ≈ e^L/19. **Right:** avalanche-size PDF on log-log axes, power law s^−1.97 (gold dashed reference). Scale-free, not exponential.

Occupied cells vs steps: early sqrt-steps growth followed by an apparent flat region — which is deceleration not saturation — **Historical figure (prior reading, now corrected).** Occupied cells vs steps: the early √steps growth and the apparent late-time plateau that led to the "saturation" claim. What looks like a plateau is severe deceleration — the frontier is still advancing but each new ring costs exponentially more steps. The live widget makes this unambiguous.

Q9 — rejection growth

Does rejection growth jam?

A different variant: place tiles only if the chosen rotation would not close a loop. If no safe rotation exists, skip the cell.

On the spiral, this never gets stuck — Q3/Q4 showed a safe rotation always exists in spiral order, even adversarially. But what about random fill order?

Under random order, rejection growth jams only at fully-enclosed holes — cells where all 6 neighbours are already placed. That configuration forces all 6 boundary edges to be interior, making every rotation close a loop. Such holes are rare (~3 × 10⁻⁴ per cell), so the density still reaches ~0.9997 before jamming completely.

The spiral's jam-free guarantee is genuinely an artifact of placement order: spiral cells are placed with at most 4 already-placed neighbours, and that geometry always leaves a safe rotation available. Random order eventually exposes the fully-enclosed configuration, but so rarely that the final density is essentially 1.

In contrast to the loop-erasing process (which actively removes tiles to maintain progress), rejection growth simply skips problematic cells. The two processes represent opposite strategies: one destructive and scale-free, the other conservative and nearly complete.

Summary

Four processes, one loop gas

All four questions ultimately reduce to properties of the underlying loop gas: the marginal exponent α ≈ 2.09 explains the glacial convergence of the loop-tile fraction; the short-range gap distribution explains why that exponent sits at its marginal value; the loop-size distribution is inherited by the avalanche spectrum (τ ≈ 1.97 ≈ α); and the geometry of spiral ordering explains why rejection growth never jams there but does in random order.

The loop-erasing process produces driven, scale-free avalanches (an SOC-like phenomenon) but with inherited rather than self-tuned criticality. The frontier advances forever at an exponentially decelerating rate — a subtler form of persistence than simple growth, and one that was initially misread as saturation.