Why does reality repeat across scales at all? The critical point, and the correlation length where the repetition stops #20

Question

At a critical point (water at 647 K and 22 MPa, a magnet at its Curie temperature), fluctuations of every size appear, from the molecular size up to a correlation length ξ ∝ |T − T_c|^−ν, which grows without limit at T_c. Can a small model, a 2D Ising lattice or percolation with seeded random numbers, reproduce the exact numbers and show the nesting stop at ξ as the temperature moves away from T_c?

Why it matters

The other studies find self-similarity over a range and ask where it stops. This one asks why it appears at all. The renormalisation group explains why scale-free behaviour comes with universal exponents that do not depend on microscopic details, and why it normally needs tuning to a special point. It is also the cleanest checkable case: the exact answers are known, and both cutoffs (the lattice spacing below; ξ or the box size above) are under the model's control. The contrast with turbulence and cosmic structure, which are scale-free over a range without tuning, belongs in the README's opening section.

First steps

• critical/model.js: Wolff or Metropolis 2D Ising on an L×L lattice with a seeded generator; magnetisation and correlation length.
• Checks: T_c = 2/ln(1+√2) = 2.269 J/k (Onsager, Phys. Rev. 65, 117 (1944)); β = 1/8 (Yang 1952); ν = 1; η = 1/4; the site-percolation threshold 0.592746 (Newman and Ziff, PRL 85, 4104 (2000)) and the incipient cluster's dimension 91/48.
• Show the range of self-similar clusters shrinking to a few lattice spacings as |T − T_c|/T_c grows. Compare real fluids through the 3D Ising exponents (ν ≈ 0.630; Kos, Poland, Simmons-Duffin and Vichi 2016).
• One README paragraph: tuned scale invariance (critical points) against self-organised ranges (turbulent cascades, gravitational clustering from a nearly scale-free initial spectrum).