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Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model

Primarily AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.MP — Mathematical Physicsmath-ph — Mathematical Physics

Contributed by P. M. Aronow ↗, Patrick Lopatto ↗claimed

A claim means that a Hexagon account has been linked to the contributor.

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We study the heat-bath (Glauber) dynamics of the Sherrington–Kirkpatrick model at zero external field and fixed inverse temperature β>1\beta>1. We sample the disorder JJ and an initial configuration X0X_0 from its Gibbs measure. Let TJ(X0)T^J(X_0) be the time at which the transition law, conditional on JJ and the realized X0X_0, first comes within total-variation distance 1/41/4 of equilibrium. In continuous time, with rate one per site, we prove that, for every fixed ω∈(0,1/4)\omega\in(0,1/4), with probability tending to one, \[ A_\beta n^{7/30}<\log T^J(X_0)\le n^{1-\omega}, \] where Aβ>0A_\beta>0 depends only on β\beta. Corresponding bounds hold for the discrete-time chain. These results strengthen OpenAI’s typical-start lower and upper bounds while following their proof architectures. The lower-bound argument introduces a three-replica comparison with a cubic pressure deficit and uses Aronow and Lopatto’s quantitative mean-bias estimate. The upper-bound argument combines a pathwise information–energy cancellation, concentration estimates on a growing mesh, and optimized exponents. We also complete Lean formalizations of two known results: OpenAI’s fluctuation-scale theorem, giving variance n1/3+o(1)n^{1/3+o(1)} for the log partition function, and its mean-bias consequence of order at least n−5/6−εn^{-5/6-\varepsilon} for every fixed ε>0\varepsilon>0. The formal development proves the positive-Laplace-window input needed for the fluctuation theorem. Five specified result groups are formalized, including the continuous-time dynamics upper bound.

Provenance statement

Proof discovery and verification were done with GPT 6 Sol, GPT 6 Astra, GPT 6.1 Sol, and Claude Opus 5.5. Lean 4 formalizations were produced by Opus 5.5 and GPT 6.1 Sol as part of verification. Final revisions to the writing were done with Claude Opus 5.5 and GPT 6.1 Sol.

References

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Version history

  1. v1Submitted by P. M. AronowInitial depositCurrentOct 20, 2026