Critical and near-critical XY scaling and BKT universality

1 · The essentials

These manuscripts study how the directions of spins on a square grid remain related across long distances. For the nearest-neighbor cosine XY model at its critical temperature, the main manuscript shows a precise rule for how that relationship weakens with distance. The rule combines a distance power with a slower logarithmic correction, going beyond a companion manuscript’s determination of the power alone. Another manuscript shows how a characteristic distance grows extremely rapidly as the critical temperature is approached from one side. Related manuscripts describe large-scale limits for random heights and spin patterns. The center-magnetization and critical spin-field conclusions depend on specified assumptions or companion results. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine rows of small arrows laid out on a tiled floor, and ask how closely the directions of two arrows remain related as you move them farther apart. The critical formula describes how that relationship fades, with a slowly changing adjustment that a simple distance-power rule would miss.

Why it matters

The main result distinguishes a precise critical correlation law from a statement giving only its leading exponent. The near-critical result addresses a different feature: the rapid growth of the correlation length on approach to the threshold. Together with the height and spin-field manuscripts, these results connect specific lattice observables with large-scale Gaussian-field descriptions, within the hypotheses stated for each model.

2 · Going further

For the square-lattice nearest-neighbor cosine XY model, let $C_{b_{c}} (r)$ denote the critical two-point correlation, with the sites separated by $r$ steps along a coordinate axis. The free-box thermodynamic limit is taken before letting $r$ grow. The main manuscript shows $C_{b_{c}} (r) = B_{XY} r^{- 1/4} (lo g r)^{1/8} (1 + o (1))$ , with $0 < B_{XY} < \infty$ . This refines the companion exponent statement $C_{b_{c}} (r) = r^{- 1/4 + o (1)}$ , which the source identifies with the BKT prediction for the critical exponent.

The near-critical companion defines a mass $m (b)$ by first taking the free-box thermodynamic limit and then the axis-separation limit. Approaching $b_{c}$ from the massive side, it shows $b_{c} - b lo g (1/ m (b)) \to A_{XY}$ , with a finite, strictly positive model-specific constant. With correlation length $ξ (b) = 1/ m (b)$ , this is the stated BKT essential singularity. The abstracts state these correlation results but do not describe the argument establishing either the logarithmic correction or the essential singularity.

The field companions give more information about their dependencies and normalization. Assuming the stated critical-height, local-renormalization, and spin-field inputs, the center magnetization in a square with aligned boundary spins is shown to be $A_{XY} n^{- 1/8} (lo g n)^{1/16} (1 + o (1))$ . Using companion critical Bessel-height and pin-limit results, the critical spin-field manuscript shows full-sequence convergence, for boundary angles fixed to zero, to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness $2/ π$ . Its normalization uses the exact center magnetization and a lattice Green-function factor; convergence is in law in $H_{loc}^{- 3} ((- 1, 1)^{2})$ , with convergence also of all mixed moments and joint laws after testing against smooth compactly supported functions.

The universality manuscript treats discrete Gaussian heights for every finite square-symmetric interaction set containing nearest neighbors. It shows Gaussian scaling limits throughout the rough phase, including its threshold; the repository summary specifies geometric torus sizes. After the natural lattice normalization, the critical effective temperature is $8 π$ . For ordinary nearest-neighbor Villain and XY models, its spin-field conclusion instead concerns sufficiently low fixed temperatures and Green-function-normalized convergence to an imaginary exponential of a Dirichlet Gaussian free field.

Useful notions

Square-lattice spin models
Two-point correlations and correlation length
Infinite-volume limits
Power laws and logarithmic corrections
Random fields and convergence in law

What was known before / what is new

Before

The source identifies the critical correlation exponent $1/4$ as a Berezinskii–Kosterlitz–Thouless prediction. A companion manuscript shows the exponent-level statement $n^{- 1/4 + o (1)}$ . The supplied text gives no broader historical account or chronology of prior results.

New

The main manuscript supplies the explicit logarithmic factor $(lo g r)^{1/8}$ and a finite, strictly positive amplitude for critical axis correlations. The near-critical companion shows the essential-singularity limit for the correlation length. Other manuscripts supply conditional critical center-magnetization asymptotics, a critical spin-field limit using companion inputs, and Gaussian height limits extending to the roughening threshold.

Limits and assumptions

The precise XY correlation statements concern the square-lattice nearest-neighbor cosine model and coordinate-axis separation, with the free-box infinite-volume limit taken first. The essential-singularity limit approaches criticality from the massive side. Critical center magnetization retains its stated height, local-renormalization, and spin-field assumptions, while the critical spin-field result uses companion Bessel-height and pin-limit results and fixed zero boundary angles. The broader height conclusion requires finite square-symmetric interactions containing nearest neighbors and is stated along geometric torus sizes; the Villain and XY spin-field conclusion in the universality manuscript is at sufficiently low fixed temperatures. The main result is not formalized in Lean and has not been formally verified. The source does not specify further open problems or give the main correlation arguments.

Mini glossary

Two-point correlation: A quantity describing how spin directions at two sites are related.
Thermodynamic limit: The infinite-volume limit obtained here by enlarging free square boxes.
Logarithmic correction: A logarithmic factor multiplying a leading power law, here $(lo g r)^{1/8}$ .
Correlation length: The characteristic length $ξ = 1/ m$ associated with the mass in the near-critical statement.
Essential singularity: The near-critical behavior expressed here by $b_{c} - b lo g ξ (b)$ tending to a strictly positive finite constant.
Gaussian free field: The Gaussian random field whose imaginary exponential appears as the limiting spin field in the companion statements.

Keywords : XY model · BKT scaling · Gaussian fields

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

24/09/2026

To be verified

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 $8 π$ . 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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