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Math-2934 · mathematics · Probability

The essential singularity of the planar XY correlation length

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
In the nearest-neighbor planar XY model below its physical inverse-temperature threshold $b_c$, does $\operatorname{sqrt}(b_c-b)$ log xi(b) converge to a finite strictly positive constant as b increases to $b_c$?

Short English statement

In the nearest-neighbor planar XY model below its physical inverse-temperature threshold $b_c$, does $\operatorname{sqrt}(b_c-b)$ log xi(b) converge to a finite strictly positive constant as b increases to $b_c$?

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Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2934-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2934

The XY correlation length is claimed to satisfy $\sqrt{b_c-b}\log(1/m(b))\to A$ with $0<A<\infty$ on approach from the massive side. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The essential singularity of the planar XY correlation length

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The XY correlation length is claimed to satisfy $\sqrt{b_c-b}\log(1/m(b))\to A$ with $0<A<\infty$ on approach from the massive side.

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. For $0<b<b_c$, the axis correlation length is $\xi(b)=m(b)^{-1}$. There exists $A_{\mathrm{XY}}\in(0,\infty)$ such that \[ \lim_{b\uparrow b_c}\sqrt{b_c-b}\,\log\xi(b)=A_{\mathrm{XY}}. \] This concerns the nearest-neighbour cosine model and approach from its positive-mass phase.

Outline of the argument

The analytic activity map from the critical companions is followed on scales that grow with the inverse-temperature distance. Dual-height fluctuation estimates identify the scale at which the marginal flow leaves the critical regime, and the spin--height mass relation converts that scale into the essential singularity.

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

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