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Math-2906 · mathematics · Probability

Critical SK mixing, relaxation and log-Sobolev exponents

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For Gaussian SK at $\beta =1$ and each fixed total-variation threshold $\epsilon \in (0,1/2)$, are continuous-time mixing, relaxation and log-Sobolev constants $n^(2/3+o(1))$, while single-site discrete mixing is $n^(5/3+o(1))$, in probability over disorder?

Short English statement

For Gaussian SK at $\beta =1$ and each fixed total-variation threshold $\epsilon \in (0,1/2)$, are continuous-time mixing, relaxation and log-Sobolev constants $n^(2/3+o(1))$, while single-site discrete mixing is $n^(5/3+o(1))$, in probability over disorder?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2906-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2906

The critical SK worst-start mixing exponent is claimed to be $2/3$ in per-site clock units, or $5/3$ in uniform single-update units. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Critical SK mixing, relaxation and log-Sobolev exponents

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The critical SK worst-start mixing exponent is claimed to be $2/3$ in per-site clock units, or $5/3$ in uniform single-update units.

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 $Q_i$ be conditional expectation given all other spins, and define $\mathcal D_J(f)=\sum_i\pi[\operatorname{Var}(f\mid x_{-i})]$. Let $t_{\mathrm{mix}}(\varepsilon)$ and $k_{\mathrm{mix}}(\varepsilon)$ be worst-start total-variation mixing times for the respective clocks. Also put \[ R_J=\sup_{\operatorname{Var}_\pi f>0}\frac{\operatorname{Var}_\pi f}{\mathcal D_J(f)},\qquad C_{\mathrm{LS},J}=\sup_{\operatorname{Ent}_\pi(f^2)>0}\frac{\operatorname{Ent}_\pi(f^2)}{\mathcal D_J(f)}, \] where $\operatorname{Ent}_\pi(g)=\pi(g\log g)-\pi(g)\log\pi(g)$. For fixed $0<\varepsilon<1/2$, in disorder probability, \[ t_{\mathrm{mix}}(\varepsilon)=n^{2/3+o(1)},\quad k_{\mathrm{mix}}(\varepsilon)=n^{5/3+o(1)},\quad R_J=n^{2/3+o(1)},\quad C_{\mathrm{LS},J}=n^{2/3+o(1)}. \] Each exponent assertion means upper and lower bounds with every fixed positive exponent error on events of probability tending to one.

Outline of the argument

Two upper-bound mechanisms combine Gaussian observations with a critical support-profile comparison. Spectral and spherical estimates in a polylogarithmic edge window control unrestricted rotation orbits; the critical slowing-down companion supplies the matching lower exponent.

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. 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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