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Dynamics of the spherical Sherrington–Kirkpatrick model, Part I: Foundations and tools

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

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

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We study the Langevin dynamics of the spherical Sherrington–Kirkpatrick model, the diffusion on the sphere of radius N\sqrt N in RN\mathbb R^N that is reversible with respect to the Gibbs measure with density proportional to exp⁡(βxTWx/2)\exp(\beta x^{\mathsf T}Wx/\sqrt2), where WW is an N×NN\times N matrix from the Gaussian orthogonal ensemble. The dynamics is started from the uniform distribution on the sphere, independently of WW. We determine the asymptotic quenched total-variation distance to equilibrium as a function of time in the fixed-temperature, mesoscopic, critical-window, and infinite-temperature crossover regimes described below, with convergence in probability or in distribution with respect to WW. Let βc=1/2\beta_c=1/\sqrt2 be the critical inverse temperature. For β<βc\beta<\beta_c there is cutoff with a Gaussian profile at fixed β∈(0,βc)\beta\in(0,\beta_c), along every mesoscopic approach of β\beta to βc\beta_c from below, and in the infinite-temperature crossover when NβN2→∞N\beta_N^2\to\infty; when NβN2N\beta_N^2 stays bounded, the distance relaxes over bounded times and there is no cutoff. At fixed β∈(0,βc)\beta\in(0,\beta_c) the mixing time is (log⁡N−(3/2)log⁡log⁡N+O(1))/(4(1−2β)2)(\log N-(3/2)\log\log N+O(1))/(4(1-\sqrt2\beta)^2), and the correction −(3/2)log⁡log⁡N-(3/2)\log\log N comes from the edge of the spectrum of WW. For β>βc\beta>\beta_c, at fixed β\beta and along every mesoscopic approach to βc\beta_c from above, there is cutoff at a time set by the gap between the two largest eigenvalues of WW, of order N2/3log⁡NN^{2/3}\log N at fixed β\beta. The profile is the total-variation distance between a standard Gaussian variable and its sum with an independent Cauchy variable; the Cauchy variable is the ratio of the first two eigencoordinates at an early stage of the evolution. In the critical window 2β=1+σN−1/3\sqrt2\beta=1+\sigma N^{-1/3}, the distance at time N2/3sN^{2/3}s converges in distribution to a random profile determined by the Airy point process, and there is no cutoff. As σ→±∞\sigma\to\pm\infty, this profile, suitably recentered and rescaled, converges to the profiles of the two mesoscopic regimes.

Provenance statement

Proof discovery and verification were done with a combination of Opus 5.5, Fable 5.1, GT 5.6 Sol, GPT 6 Luna, GPT 6 Sol, GPT 6 Astra, GPT 6.1 Terra, and GPT 6.1 Sol. The results were formalized in Lean 4 by Opus 5.5, Luna 6, Terra 6.1, and Sol 6.1. The writing was done by Opus 5.5, Sol 5.6, Sol 6.1, and Astra 6.

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  1. v1Submitted by Patrick LopattoInitial depositCurrentOct 28, 2026