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Critical and near-critical XY scaling and BKT universality

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For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations \(C_{\beta_c}(r)\sim Ar^{-1/4}(\log r)^{1/8}\) and the Berezinskii–Kosterlitz–Thouless essential singularity \(\sqrt{\beta_c-\beta}\log\xi(\beta)\to B\), with \(A,B\gt 0\) after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1 (Height universality through the roughening threshold)

In BKT universality for height and planar spin fields

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Eight other possible matches are listed below the card, among them Theorem 1.2 (Full low-temperature spin-field limits) in the same paper and Theorem 1.1 in The Critical Spin Field of the Planar XY Model.

For every admissible J, the threshold in Equation (1.3) satisfies 0<\bc(J)<\infty. The supremum there is finite when 0<\beta<\bc(J) and infinite when \beta\ge\bc(J), including equality.

There are an integer L=L(J)\ge2 and a unique positive function \beff(J,\beta), defined for \beta\ge\bc(J), such that along the full sequence N\to\infty

H_N\ \Longrightarrow\ \sqrt{\frac{\beff(J,\beta)}{v_J^2}}\,\Phi_{\T} \quad\hbox{in }H^{-3}(\T^2).

For every smooth real mean-zero f,

\lim_{N\to\infty}\log\E e^{H_N(f)} =\frac{\beff(J,\beta)}{2v_J^2} (f,(-\Delta_{\T})^{-1}f).

Furthermore,

\begin{aligned}\beff(J,\bc(J))&=8\pi v_J^2,\\ \lim_{\beta\downarrow\bc(J)} \frac{\beff(J,\beta)-\beff(J,\bc(J))}{\beta-\bc(J)} &=+\infty.\end{aligned}

The function \beff(J,\cdot) is differentiable on (\bc(J),\infty), and for some c_J>0,

\beff(J,\beta)=\beta+O_J(e^{-c_J\beta})\qquad(\beta\to\infty).

For J_\rho=\{x\in\Z^2\setminus\{0\}:|x|_\infty\le\rho\},

\frac{\bc(J_\rho)}{8\pi v_{J_\rho}^2}\longrightarrow1 \qquad(\rho\to\infty).

In particular v_{J_{\nn}}^2=1/4 and the nearest-neighbor effective temperature at roughening is 2\pi.

Eight more possible matches

  • Theorem 1.2 (Full low-temperature spin-field limits) in BKT universality for height and planar spin fields.
  • Theorem 1.1 in The Critical Spin Field of the Planar XY Model.
  • Theorem 1.1 in Critical Center Magnetization in the Planar XY Model.
  • Theorem 1.1 in Essential Singularity of the Correlation Length in the Planar XY Model, 5 October 2026.
  • Theorem 1.1 in The critical correlation exponent of the planar XY model.
  • Theorem 1.3 (Villain coefficient bridge) in BKT universality for height and planar spin fields.
  • Proposition 1.2 (Critical polynomial lower bound) in The critical correlation exponent of the planar XY model.
  • Theorem 1.1 in The critical logarithmic correction for the planar XY model, 5 October 2026.
Reported proof verification
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Papers and explanations
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Papers

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The critical logarithmic correction for the planar XY model

5 October 2026, current edition.

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We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let \(C_{b_c}(r)\) be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then \(\displaystyle C_{b_c}(r)=B_{\mathrm{XY}}r^{-1/4}(\log r)^{1/8}(1+o(1)), \qquad B_{\mathrm{XY}}\in(0,\infty),\) as \(r\to\infty\).

Critical Center Magnetization in the Planar XY Model

Undated, current edition.

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Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As \(n\to\infty\), the magnetization is \(A_{\mathrm{XY}}n^{-1/8}(\log n)^{1/16}(1+o(1))\), where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

The Critical Spin Field of the Planar XY Model

Undated, current edition.

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Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness \(2/\pi\). The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in \(H^{-3}_{\mathrm{loc}}((-1,1)^2)\); all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in \((-1,1)^2\) converge as well.

Essential Singularity of the Correlation Length in the Planar XY Model

5 October 2026, current edition.

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We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let \(m(b)\) be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches bc from the massive side, \(\displaystyle \sqrt{b_c-b}\log\frac{1}{m(b)} \longrightarrow A_{\mathrm{XY}},\) where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

The critical correlation exponent of the planar XY model

Undated, current edition.

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For the ordinary nearest-neighbor XY model on the square lattice, we prove the Berezinskii–Kosterlitz–Thouless prediction that the critical spin-correlation exponent is 1/4. More precisely, at the mass-defined critical inverse temperature, the infinite-volume correlation is \(n^{-1/4+o(1)}\). The infinite-volume limit through free square boxes is taken before the separation limit.

BKT universality for height and planar spin fields

Undated, current edition.

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We prove Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the physical roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbors. After the natural lattice normalization, the critical effective temperature has the universal value \(8\pi\). For the ordinary nearest-neighbor Villain and XY models at sufficiently low fixed temperatures, we also prove that their Green-function-normalized spin fields converge to the imaginary exponential of a Dirichlet Gaussian free field.

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  • seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source

    Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.

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