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Math-2910 · mathematics · Probability

A size-independent Gibbs Poincaré constant throughout high-temperature SK

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed $0\leq \beta <1$ in Gaussian zero-field SK, is there $C_\beta <infinity$ such that, with probability tending to one over disorder, $Var_\mu (f)\leq C_\beta $ D(f) simultaneously for every spin function? Equivalently, the single-site discrete heat-bath spectral gap is at least $1/(C_\beta $ n).

Short English statement

For every fixed $0\leq \beta <1$ in Gaussian zero-field SK, is there $C_\beta <infinity$ such that, with probability tending to one over disorder, $Var_\mu (f)\leq C_\beta $ D(f) simultaneously for every spin function? Equivalently, the single-site discrete heat-bath spectral gap is at least $1/(C_\beta $ n).

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2910-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2910

Throughout fixed SK temperatures $0<\beta<1$, the rate-one heat-bath gap is claimed bounded below by a positive constant with high disorder probability. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A size-independent Gibbs Poincaré constant throughout high-temperature SK

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Throughout fixed SK temperatures $0<\beta<1$, the rate-one heat-bath gap is claimed bounded below by a positive constant with high disorder probability.

Mathematical statement of the claimed result

Result 1.
For independent standard Gaussian $J_{ij}$, $i<j$, with $J_{ji}=J_{ij},J_{ii}=0$, use the zero-field SK measure \[ \pi_{n,\beta}^J(x)=Z^{-1}\exp\left(\frac\beta{\sqrt n}\sum_{i<j}J_{ij}x_ix_j\right),\qquad x\in\{-1,1\}^n. \] Continuous heat-bath dynamics refreshes every site at rate one; a discrete step refreshes one uniformly selected site, including attempts that retain its spin. For fixed $0<\beta<1$, a finite deterministic $C_\beta$ satisfies \[ \mathbb P_J\left\{\operatorname{Var}_\pi(f)\le C_\beta\sum_{i=1}^n \mathbb E_\pi(f-\mathbb E_\pi[f\mid x_{-i}])^2 \text{ for all }f:\{-1,1\}^n\to\mathbb R\right\}\longrightarrow1. \] The continuous-time spectral gap is at least $C_\beta^{-1}$ on this event; the uniform-site discrete gap is at least $(nC_\beta)^{-1}$. This theorem uses zero external field.

Outline of the argument

Gaussian-observation stochastic localization transports a functional inequality from posterior systems. Stopped entropy estimates control rare observation paths weighted by a test function, and stable-field plus random-matrix estimates retain enough variance to obtain a dimension-free Poincare bound.

Formal verification

For family 227, the selected statements include the high-temperature gap for $\beta<1$, cutoff ratios only for $\beta<1/2$, critical lower mixing scales with a subexponential upper bound, and low-temperature barriers. They do not provide the full critical exponent, the full cutoff location, or the newer autocorrelation limits. The listed Comparator files are CriticalSK.lean, CriticalSKMixing.lean, SKBarriers.lean, SKHighTemperature.lean, SKRatio.lean. Named declarations include OAI.CriticalSK.covariance_and_linear_tests, OAI.CriticalSK.critical_mixing_bounds. 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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