Fast mixing of Glauber dynamics for the Sherrington-Kirkpatrick model throughout the high-temperature phase, zero external field

Spectral-gap paper, Theorem 1.1: for fixed $0<\beta<1$ , with probability tending to one over the disorder, $\mathrm{Var}_{\mu_0}(f)\le C_\beta\mathcal D_0(f)$ for all $f$ (zero field), so the uniform-site heat-bath chain has gap at least $1/(C_\beta n)$ . Companions: worst-start total-variation cutoff at $\log n/(2\lambda(\beta))$ for $0\le\beta<1$ ; at $\beta=1$ mixing time $n^{2/3+o(1)}$ and slow mixing from typical starts before $n^{2/3}$ ; for $\beta>1$ stretched-exponential barriers from typical starts and an $\exp(n^{1-c})$ typical-start upper bound; at $\beta=1$ universal random limits of rescaled autocorrelations for Gaussian and Rademacher disorder. Not shown: nonzero external fields, or an $O(n\log n)$ bound from the gap paper alone.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Fast mixing of Glauber dynamics for the Sherrington-Kirkpatrick model throughout the high-temperature phase, zero external field · SK Glauber dynamics spectral gap for all beta < 1

Confidence: Not scored

Registry verification: lean checked · announcement · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Fast mixing of Glauber dynamics for the Sherrington-Kirkpatrick model throughout the high-temperature phase, zero external field

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the spectral-gap paper were read against Open Problem 15. Scope limit: zero external field only, while the problem asks for arbitrary fields; polynomial mixing follows from the gap by the standard bound (about $n^2$ updates), not stated as a separate theorem there. Lean: the challenge SKHighTemperature (theorem OAI.SKGap.sk_main, solution module OAI.Probability.SKGap.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here: for each $0<\beta<1$ a constant $C$ such that the zero-field Poincare inequality with the heat-bath Dirichlet form, and discrete gap at least $1/(Cn)$ , hold with disorder probability tending to one. That is the principal theorem. Not rebuilt here. Other Lean challenges (CriticalSK, CriticalSKMixing, SKRatio, SKBarriers) cover companion results; SKRatio covers only beta < 1/2.

Source ↗

Fast mixing of Glauber dynamics for the Sherrington-Kirkpatrick model throughout the high-temperature phase, zero external field

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Companion: Cutoff throughout the high-temperature Sherrington-Kirkpatrick phase

paper · pending