Conformal universality for weakly interacting and random-bond Ising models

What OpenAI claims

Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE3 interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments. (Lean)

Papers (6)

1. Quenched SLE₃ limits for general weak random-bond Ising models

   5 October 2026 · Preprint, not peer reviewed · PDF

   For the planar Ising model with bonds J_e=1+εξ_e, where the ξe have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE3 convergence for sufficiently small ε > 0. The temperature is the spontaneous-magnetization threshold. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

2. Quenched SLE₃ Universality for the Weak Random-Bond Ising Model

   5 October 2026 · Preprint, not peer reviewed · PDF

   We prove quenched chordal SLE3 convergence for the planar Ising model with sufficiently weak independent symmetric two-valued ferromagnetic bonds. The temperature is the critical point defined by spontaneous magnetization. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

3. Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model

   5 October 2026 · Preprint, not peer reviewed · PDF

   Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds 1±ε grows as (log r)^1/4+o(1) for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.

4. Conformal universality of bulk Ising correlations under weak interactions

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign. One critical-temperature branch and two field normalizations apply in every bounded simply connected C2 Jordan domain with free, plus, or minus boundary conditions, and in the periodic thermodynamic plane state. Energies are centered in their actual states.

5. SLE3 universality for weak finite-range Ising interactions

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE3. The result allows interactions of either sign, arbitrary admissible exterior contours, and uniformly approximated Jordan domains. Convergence holds for the full oriented curve law in uniform distance modulo increasing reparametrization.

6. Buffered comparison and stopping-band resolution in critical Ising

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted. For an interface exploration across a band of fixed positive width, finitely many regular signed barriers approximate every good conditional target law in total variation, uniformly over bounded observables and sufficiently fine meshes.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.