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.
Manuscripts (6)
The critical logarithmic correction for the planar XY model
October 5, 2026 · source · cite as \cite{OAI:The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026}
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
October 5, 2026 · source · cite as \cite{OAI:Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026}
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
October 5, 2026 · source · cite as \cite{OAI:The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026}
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
October 5, 2026 · source · cite as \cite{OAI:Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026}
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
September 24, 2026 · source · cite as \cite{OAI:The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026}
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
September 24, 2026 · source · cite as \cite{OAI:BKT-universality-for-height-and-planar-spin-fields-September-24-2026}
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.
Result 216 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md