Spectral gaps of high-temperature SK heat-bath dynamics
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For the zero-field Gaussian Sherrington–Kirkpatrick model at each fixed 0<β<1, seek a dimension-independent Poincaré bound for all observables for rate-one-per-site heat-bath dynamics, with disorder probability tending to one. Equivalently, uniformly selected single-site updates should have gap at least 1/(Cβ n).
Why it matters and what counts as progress
Why it matters: It asks for control of the slowest heat-bath mode throughout the high-temperature phase, beyond covariance bounds for linear observables.
What counts as progress: Extend a dimension-independent all-observable Poincaré bound through the full fixed β<1 range; covariance estimates alone do not suffice.
How the problem developed
- 16 Jul 2020 · Partial result [1]
Ronen Eldan, Frederic Koehler and Ofer Zeitouni Proved a dimension-free Poincaré inequality for Ising measures when the positive interaction matrix has operator norm below one; this yielded a smaller high-temperature range for SK dynamics.
- 24 Sept 2026 · Proof claim [2]
OpenAI Reported a dimension-independent Poincaré inequality for zero-field Gaussian SK heat-bath dynamics at every fixed 0<β<1, with disorder probability tending to one.
AI contribution
A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase
For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed 0<β<1, rate-one-per-site heat-bath dynamics satisfy a dimension-independent Poincaré inequality for all observables with disorder probability tending to one. Uniformly chosen single-site updates have gap at least 1/(Cβ n).
Sources
- Eldan, Koehler and Zeitouni, A spectral condition for spectral gap: fast mixing in high-temperature Ising models
- OpenAI high-temperature SK spectral-gap manuscript
- A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase
- OpenAI SK spectral-gap formalization scope
- OpenAI SK spectral-gap Lean theorem
- OpenAI mathematics release account
Website timeline
Publication dates and editorial changes.
OpenAI reports a spectral gap across the high-temperature SK phase