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All-temperature correlation decay in two-dimensional O(n) models

For every integer n≥3 and finite β>0, do spin two-point correlations in finite free-boundary nearest-neighbour square-lattice O(n) models with edge strengths in [0,β] satisfy a uniform A exp(−m|x−y|) bound? The target also passes to free-boundary limits.

Why it matters and what counts as progress

Why it matters: It asks for exponential decay throughout the stated O(n) models at every finite temperature, rather than only weak correlation or high-temperature bounds.

What counts as progress: Prove uniform all-temperature exponential spin-correlation decay for n≥3, with boundary conditions and finite-volume constants controlled.

How the problem developed

  1. 1975 · Theoretical prediction [1][2]

    Alexander M. Polyakov Developed the low-temperature perturbative picture for two-dimensional non-Abelian O(n) models that later motivated the expected exponential decay of correlations.

  2. 11 Mar 1999 · Open problem [2]

    Adrian Patrascioiu and Erhard Seiler Recorded the question whether classical two-dimensional Heisenberg O(n>2) correlations decay exponentially at all nonzero temperatures; the page noted that no proof existed then.

  3. 23 Sept 2026 · Reported claim [3]

    OpenAI The preprint claims exponential correlation decay at every finite temperature for the stated two-dimensional classical O(n) models with n≥3.

AI contribution

Sources

  1. Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields
  2. Soft phases in two-dimensional O(N) models
  3. OpenAI 2D O(n) model manuscript
  4. Exponential decay in two-dimensional classical O(n) models
  5. Lean spin-correlation decay theorem
  6. Formalization scope for the O(n) result
  7. OpenAI mathematics release account
  8. Earlier open-problem discussion
Website timeline

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