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Overlap distribution of the critical Sherrington-Kirkpatrick model — arXiv:2608.08752

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.08752
License (as archived): CC0-1.0
Archived: 2026-10-19T09:07:00.345Z
SHA-256 of the archived text: 21d6caf6c49d7ae7ea9fa073901efa53f474c8055781544b12d6c725c4b4cf08
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Overlap distribution of the critical Sherrington-Kirkpatrick model — arXiv:2608.08752 — arXiv (en), https://arxiv.org/abs/2608.08752 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_b166cb68dc41 (Metraipolis archive dc_b166cb68dc41, sha256 21d6caf6c49d7ae7ea9fa073901efa53f474c8055781544b12d6c725c4b4cf08)

BibTeX

@misc{dcb166cb68dc41,
  title = {Overlap distribution of the critical Sherrington-Kirkpatrick model — arXiv:2608.08752},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.08752},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_b166cb68dc41; sha256 21d6caf6c49d7ae7ea9fa073901efa53f474c8055781544b12d6c725c4b4cf08},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.08752. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the distribution of the two-replica overlap $R_{1,2}$ in the Ising and spherical Sherrington-Kirkpatrick models at the critical inverse temperature $\beta = 1$. Our main result shows that in both models, $R_{1,2}$ has scale $N^{-1/3}$, and the quenched distribution of $N^{1/3} R_{1,2}$ converges to an explicit random probability measure defined in terms of the reflected $\mathrm{Airy}_1$ point process. As a consequence, we characterize the limiting value of $N^{2/3} \mathbb{E} \langle R_{1,2}^2 \rangle$, answering a question of Talagrand \cite{talagrand2011mean2}. For the spherical SK model, we obtain the limit by representing the Gibbs measure as an anisotropic Gaussian on $\mathbb{R}^N$ conditioned to have norm $\sqrt{N}$, and then passing to the $\mathrm{Airy}_1$ scaling limit at the GOE spectral edge. For the Ising SK model, the proof is based on a sphere-to-cube comparison principle showing that the quenched distributions of $N^{1/3} R_{1,2}$ under the spherical and Ising Gibbs measures asymptotically coincide. This paper is a companion to \cite{du2026fluctuations}, where we introduced a related comparison principle to identify the limiting fluctuations of the SK free energy. Most of the arguments in this paper were generated using GPT-5.6 Pro, with the aim of exploring further consequences of the ideas developed in that work.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-11 in math.PR).
Categorie arXiv: Probability (math.PR) ; Disordered Systems and Neural Networks (cond-mat.dis-nn); Mathematical Physics (math-ph).
Pagina del lavoro: https://arxiv.org/abs/2608.08752 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T09:04:32.176Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-11?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).