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Math-2912 · mathematics · Probability

The one-sixth free-energy fluctuation exponent in low-temperature SK

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed $\beta >1$ in Gaussian zero-field SK, are $sd(F_n)=n^(1/6+o(1))$ and $Var(F_n)=n^(1/3+o(1))$? Require additionally that $|F_n-E$ $F_n|$ lies between $n^{1/6-\eta }$ and $n^{1/6+\eta }$ with probability tending to one for every $\eta >0$.

Short English statement

For each fixed $\beta >1$ in Gaussian zero-field SK, are $sd(F_n)=n^(1/6+o(1))$ and $Var(F_n)=n^(1/3+o(1))$? Require additionally that $|F_n-E$ $F_n|$ lies between $n^{1/6-\eta }$ and $n^{1/6+\eta }$ with probability tending to one for every $\eta >0$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2912-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2912

At each fixed zero-field Gaussian SK inverse temperature $\beta>1$, both standard deviation and typical centered fluctuations are claimed to have exponent $1/6$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The one-sixth free-energy fluctuation exponent in low-temperature SK

OpenAI's claim

OpenAI presents a proof in the selected manuscript. At each fixed zero-field Gaussian SK inverse temperature $\beta>1$, both standard deviation and typical centered fluctuations are claimed to have exponent $1/6$.

Mathematical statement of the claimed result

Result 1.
For fixed $\beta>1$, let $g_{ij}$ be independent standard Gaussians and define \[ F_n=\log\left(2^{-n}\sum_{\sigma\in\{-1,1\}^n} \exp\left(\frac\beta{\sqrt n}\sum_{i<j}g_{ij}\sigma_i\sigma_j\right)\right). \] The claim is \[ \operatorname{sd}(F_n)=n^{1/6+o(1)},\qquad \operatorname{Var}(F_n)=n^{1/3+o(1)}. \] More precisely, for every $\eta>0$ and sufficiently large $n$, $n^{1/6-\eta}\le\operatorname{sd}(F_n)\le n^{1/6+\eta}$, and \[ \mathbb P\{n^{1/6-\eta}\le|F_n-\mathbb EF_n|\le n^{1/6+\eta}\}\to1. \] Probability is over the zero-field Gaussian disorder.

Outline of the argument

Gaussian covariances are revealed along finite hierarchical paths, with pinned inverse-Hessian estimates controlling moments. Optimized local comparisons and signed replica attachments cancel the susceptibility correction; the resulting small-mass overlap profile gives a positive Laplace window at scale $n^{-1/6}$ and the matching fluctuation exponent.

Formal verification

Family 217 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.

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