Family 216Probability and statistical mechanicsNo LeanNot yet digested
Critical and near-critical XY scaling and BKT universality
For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations and the Berezinskii–Kosterlitz–Thouless essential singularity , with 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 in this family · 6
The critical logarithmic correction for the planar XY model
Oct 5, 2026·paper.pdf
We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let 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
as .
Critical Center Magnetization in the Planar XY Model
Oct 5, 2026·paper.pdf
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 , the magnetization is , where is a finite, strictly positive model-specific constant.
The Critical Spin Field of the Planar XY Model
Oct 5, 2026·paper.pdf
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 . The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in ; all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in converge as well.
Essential Singularity of the Correlation Length in the Planar XY Model
Oct 5, 2026·paper.pdf
We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let 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 b from the massive side,
where is a finite, strictly positive model-specific constant.
The critical correlation exponent of the planar XY model
Sep 24, 2026·paper.pdf
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 . The infinite-volume limit through free square boxes is taken before the separation limit.
BKT universality for height and planar spin fields
Sep 24, 2026·paper.pdf
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 . 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.