Conformal universality for weakly interacting and random-bond Ising models

The explanation

Colour each square of a fine grid black or white. Make the left half of the border black and the right half white. Somewhere inside, a curve must separate black from white. In the Ising model, each square holds a spin, a tiny magnet pointing up or down. Neighbouring spins prefer to agree. The coupling, the strength of that preference times the inverse temperature, decides how strongly. At one special coupling, ½ log(1 + √2), about 0.4407, the model is critical. There the curve wiggles at every scale.

At that point the curve has an exact limit as the squares shrink. It is chordal SLE₃, a random curve driven by √3 times a Brownian motionSource 1. Chelkak, Duminil-Copin, Hongler, Kemppainen and Smirnov proved this convergence for the plain square gridSource 2. Their proof leans on the model being exactly solvable. Exactly solvable means special formulas, built from fermionic observables, describe it perfectly.

Physicists expect the limit to be universal. Universal means small changes to the microscopic rules should not change the large-scale shape. Earlier rigorous work showed this for some correlation numbers under weak interactions. It did not reach the whole curve. Greenblatt and Peltola set up one route to the curve but did not prove convergenceSource 3.

The first family of papers changes the energy. It adds small extra interactions of finite range, among any even number of spins, of either sign. They must respect the square's symmetries. One paper claims that, at a suitably shifted critical temperature, the interface still converges to SLE₃Source 4. Another claims the same universality for bulk spin and energy correlations, which measure how distant spins and bonds move togetherSource 5.

The second family makes the bonds random. Each bond gets strength 1 + εξ, where ξ is an independent random number with mean zero, bounded by 1Source 6. With ε = 0.1, a fair coin would give bonds of 1.1 or 0.9. The environment is drawn once and then held fixed. That is called quenched. One paper claims SLE₃ convergence for fair two-valued bondsSource 7. Another extends it to every bounded, mean-zero law, for ε small enoughSource 8. It also places the critical coupling within a constant times ε² of the clean valueSource 9.

The obstacle is that two-dimensional disorder is marginal. Weak disorder shrinks under zooming out, but only logarithmically. So a proof needs a decay estimate scale by scale, not one uniform contractionSource 10. A further paper measures how slowly the disorder fades. It finds that a relative second moment of spin correlations grows like (log r)^(1/4)Source 11. At distance r of one million, (log r)^(1/4) is only about 1.93. That paper is conditional. It assumes certain stated estimates for the clean critical modelSource 12.

The last paper supplies tools. It compares Ising probabilities when some spins are pinned and the boundary changesSource 13. That finite comparison is the piece a computer has checked.

How sure is this? A computer proof checker called Lean has checked a formal version of one finite-graph comparison. It bounds the likelihood ratio between two pinned Ising laws. The same paper's stopping-band approximation is outside that check. The SLE₃ limits, the bulk universality and the disorder results are not formalized. No specialist review has been published.

What it doesn't show. This does not cover strong perturbations, or disorder strong enough to change the critical behaviour.