we-solve (beta)

Math-2936 · mathematics · Probability

The critical correlation power of the nearest-neighbor planar XY model

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, does log $C_{b_c}(r)/\log $ r tend to $-1/4$ along integer axial separations r$\to$infinity? This question asks only the power, without an amplitude limit or prescribed logarithmic factor.

Short English statement

For the nearest-neighbor planar XY model at its physical critical inverse temperature, does log $C_{b_c}(r)/\log $ r tend to $-1/4$ along integer axial separations r$\to$infinity? This question asks only the power, without an amplitude limit or prescribed logarithmic factor.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2936-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2936

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 question retains these same parameters and conventions. The positive-amplitude asymptotic C(r)~B r^(-1/4)(log r)^(1/8) implies the requested logarithmic slope and gives strictly more information. The claimed conclusion supplies the requested case and extends it in the directions just specified.

The critical correlation power of the nearest-neighbor planar XY model

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

Go to contribution →