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Math-2908 · mathematics · Probability

Critical SK mixing lower bounds from realized Gibbs states

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
At Gaussian SK $\beta =1$, does the Gibbs probability of initial states whose continuous-time total-variation distance exceeds 1/4 at any deterministic $t_n=o(n^{2/3})$ tend to one in disorder probability? Ask for the same claim at $k_n=o(n^{5/3})$ single-site discrete updates.

Short English statement

At Gaussian SK $\beta =1$, does the Gibbs probability of initial states whose continuous-time total-variation distance exceeds 1/4 at any deterministic $t_n=o(n^{2/3})$ tend to one in disorder probability? Ask for the same claim at $k_n=o(n^{5/3})$ single-site discrete updates.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2908-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2908

At SK criticality, typical fixed equilibrium starts are claimed to remain far from equilibrium for every time $o(n^{2/3})$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Critical SK mixing lower bounds from realized Gibbs states

OpenAI's claim

OpenAI presents a proof in the selected manuscript. At SK criticality, typical fixed equilibrium starts are claimed to remain far from equilibrium for every time $o(n^{2/3})$.

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. At $\beta=1$, let $d_t^{\mathrm{ct}}(J,x)$ and $d_k^{\mathrm{dt}}(J,x)$ be the transition-law total-variation distances from $\pi$ for a fixed realized initial state $x$. Every deterministic sequence $t_n=o(n^{2/3})$ and integer sequence $k_n=o(n^{5/3})$ satisfy \[ \pi\{x:d_{t_n}^{\mathrm{ct}}(J,x)>1/4\}\xrightarrow{\mathbb P}1,\qquad \pi\{x:d_{k_n}^{\mathrm{dt}}(J,x)>1/4\}\xrightarrow{\mathbb P}1. \] Probability here concerns the Gaussian interactions; the Gibbs sample is fixed before evaluating its distance.

Outline of the argument

Edge-scale resolvent traces and Gaussian conditioning produce a slow critical projection of a Gibbs sample. Overlap and dynamical comparison estimates show that this projection cannot decorrelate on a smaller timescale, giving a total-variation obstruction for most sampled starting states.

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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