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Math-2935 · mathematics · Probability

The logarithmic factor in critical planar XY correlations

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the nearest-neighbor planar XY model at its physical critical inverse temperature $b_c$, is its axial two-spin correlation $C_{b_c}(r)$ asymptotic to B $r^{-1/4}(\log $ $r)^{1/8}$, for an amplitude $B>0$ and integer $r\to \infty $?

Short English statement

For the nearest-neighbor planar XY model at its physical critical inverse temperature $b_c$, is its axial two-spin correlation $C_{b_c}(r)$ asymptotic to B $r^{-1/4}(\log $ $r)^{1/8}$, for an amplitude $B>0$ and integer $r\to \infty $?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2935-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2935

The logarithmic-correction manuscript claims a positive amplitude and the complete asymptotic $B r^{-1/4}(\log r)^{1/8}(1+o(1))$ requested here. The earlier exponent manuscript supplies only the logarithmic slope $-1/4$. The full match uses the later asymptotic, while the earlier result alone would not identify either the amplitude or the logarithmic factor.

The logarithmic factor in critical planar XY correlations

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: At the ordinary XY critical point, the proposed infinite-volume two-point law has exponent $1/4$, with the free-box limit taken first; The critical XY correlation is claimed to be $B r^{-1/4}(\log r)^{1/8}(1+o(1))$ with $B>0$. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
For angles $\theta_x\in\mathbb R/(2\pi\mathbb Z)$ on $\Lambda_R=[-R,R]^2\cap\mathbb Z^2$, use the free-boundary cosine law proportional to $\exp[b\sum_{\{x,y\}\subset\Lambda_R,\,|x-y|=1}\cos(\theta_x-\theta_y)]$, relative to product uniform angles. Count each edge once and omit exterior edges. Define \[ C_b(r)=\lim_{R\to\infty}\mathbb E_{R,b}^{\mathrm{free}}\cos(\theta_0-\theta_{re_1}),\quad m(b)=\lim_{r\to\infty}-r^{-1}\log C_b(r),\quad b_c=\inf\{b>0:m(b)=0\}. \] The thermodynamic limit precedes the separation limit. At the critical point, \[ \lim_{n\to\infty}\frac{\log C_{b_c}(n)}{\log n}=-\frac14. \] Equivalently, for every $\varepsilon>0$ and all sufficiently large integers $n$, $n^{-1/4-\varepsilon}\le C_{b_c}(n)\le n^{-1/4+\varepsilon}$. This paper determines the power exponent; a separate selected manuscript supplies the logarithmic factor.
Result 2.
For angles $\theta_x\in\mathbb R/(2\pi\mathbb Z)$ on $\Lambda_R=[-R,R]^2\cap\mathbb Z^2$, use the free-boundary cosine law proportional to $\exp[b\sum_{\{x,y\}\subset\Lambda_R,\,|x-y|=1}\cos(\theta_x-\theta_y)]$, relative to product uniform angles. Count each edge once and omit exterior edges. Define \[ C_b(r)=\lim_{R\to\infty}\mathbb E_{R,b}^{\mathrm{free}}\cos(\theta_0-\theta_{re_1}),\quad m(b)=\lim_{r\to\infty}-r^{-1}\log C_b(r),\quad b_c=\inf\{b>0:m(b)=0\}. \] The thermodynamic limit precedes the separation limit. There exists a finite positive $B_{\mathrm{XY}}$ satisfying \[ C_{b_c}(r)=B_{\mathrm{XY}}r^{-1/4}(\log r)^{1/8}(1+o(1))\quad(r\to\infty\text{ through positive integers}). \] This fixes the logarithmic multiplier as well as the critical power.

Outline of the argument

The dual Bessel-height model identifies the physical mass threshold and its critical stiffness. Multiscale height estimates, combined with the exact spin--height correspondence, translate that stiffness into the logarithmic slope of the spin correlation.

The companion critical-height input fixes the leading stiffness, and an analytic activity map tracks its marginal running correction. Summing the renormalization increments gives the logarithmic factor and controls the remaining amplitude in the thermodynamic two-point function.

Formal verification

Family 216 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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