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Math-2931 · mathematics · Probability

Logarithmic relative fluctuations of disordered critical Ising correlations

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assuming the deterministic reference-Ising bounds listed in the source, does the relative quenched two-spin fluctuation $R_\epsilon (r)$ of the planar two-valued weak random-bond model satisfy log $R_\epsilon (r)/\log $ log $r\to 1/4$ for every sufficiently small fixed $\epsilon >0$? Use the physical critical beta solving $sinh(2\beta (1-\epsilon ))sinh(2\beta (1+\epsilon ))=1$.

Short English statement

Assuming the deterministic reference-Ising bounds listed in the source, does the relative quenched two-spin fluctuation $R_\epsilon (r)$ of the planar two-valued weak random-bond model satisfy log $R_\epsilon (r)/\log $ log $r\to 1/4$ for every sufficiently small fixed $\epsilon >0$? Use the physical critical beta solving $sinh(2\beta (1-\epsilon ))sinh(2\beta (1+\epsilon ))=1$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2931-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2931

At weak binary disorder, the relative second moment of critical Ising spin correlation is claimed to grow like $(\log r)^{1/4+o(1)}$, assuming the listed reference estimates. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Logarithmic relative fluctuations of disordered critical Ising correlations

OpenAI's claim

OpenAI presents a proof in the selected manuscript. At weak binary disorder, the relative second moment of critical Ising spin correlation is claimed to grow like $(\log r)^{1/4+o(1)}$, assuming the listed reference estimates.

Mathematical statement of the claimed result

Result 1.
On square-lattice bonds let $J_e=1+\varepsilon\xi_e$, with independent fair signs $\xi_e$. Define the physical critical inverse temperature by \[ \beta_c(\varepsilon)=\inf\left\{\beta\ge0: \mathbb E_{\mathrm{env}}\lim_{m\to\infty}\langle\sigma_0\rangle^+_{[-m,m]^2,\beta,J}>0\right\}, \] normalizing each Gibbs law before environmental averaging. Let $C_{\varepsilon,\omega}(r)=\lim_{n\to\infty}\langle\sigma_0\sigma_{(r,0)}\rangle^{\mathrm{free}}_{[-n,n]^2,\beta_c(\varepsilon),\omega}$ and \[ R_\varepsilon(r)=\frac{\mathbb E C_{\varepsilon,\omega}(r)^2}{(\mathbb E C_{\varepsilon,\omega}(r))^2}. \] The theorem takes deterministic critical-reference correlation, conditional-averaging and marked-source estimates as inputs. Its conditional-averaging input can be expressed with $P_RF=\mathbb E[F\mid\text{cutting spins of }[-R,R]^2]$, $F^\circ=F-\langle F\rangle_0$ and $d_R(F)=\langle F\rangle^+_{\mathrm{disk}(4R)}-\langle F\rangle_0$: \[ |d_R(F)|\le C(w/R)^b\|F\|_\infty,quad \|P_RF^\circ\|_\infty\le C|d_R(F)|+C(w/R)^p\|F\|_\infty, \] for $F$ supported in a radius-$w$ square, sufficiently large fixed $R/w$, and fine scales. The pairs $(b,p)$ are $(1,3)$ for even $F$ invariant under right-angle rotations, $(1,2)$ for other even $F$, and $(1/8,9/8)$ for odd $F$. The energy and spin calibrations have nonzero limits $R^bd_R(\mathcal O_b)$, where $\mathcal O_1$ is the average of four centred incident bond products and $\mathcal O_{1/8}=\sigma_0$. Subject to these deterministic inputs and their bulk field-product estimates, some $\varepsilon_0>0$ gives, for fixed $0<\varepsilon<\varepsilon_0$, \[ \lim_{r\to\infty}\frac{\log R_\varepsilon(r)}{\log\log r}=\frac14. \] The physical critical temperature is the unique positive root of \[ \sinh(2\beta(1-\varepsilon))\sinh(2\beta(1+\varepsilon))=1. \] The deterministic calibration input also requires, for $b\in\{1,1/8\}$, \[ c\le\|w^bP_w\mathcal O_b\|_\infty\le C,\qquad c(w/R)^b\le|d_R(w^bP_w\mathcal O_b)|\le C(w/R)^b, \] with the same bulk scale and buffer conventions as the operator estimates.

Outline of the argument

The proof retains the separate quenched Gibbs normalizations when forming disorder moments. Critical-reference comparisons control stochastic bond increments and locate the physical critical point; cumulative marginal corrections determine the logarithmic second-moment growth.

Formal verification

For family 218, the formal scope is a finite-graph likelihood comparison through the zero-field FK connection probability after common pins are deleted. Stopping-band resolution and the universality companions are excluded. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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.

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