A quantitative replica-symmetric bound for Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region
Abstract
We consider the Sherrington--Kirkpatrick model with inverse temperature β>0 and deterministic external field h>0. Let q be the replica-symmetric fixed point: q=𝔼tanh2 (h+β√q Z), where Z is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida--Thouless region: β2𝔼sech4(h+β√q Z) <1, the overlap satisfies the concentration: 𝔼⟨ (R12−q)2⟩=O(N−1). As a consequence, we obtain an O(N−1) replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. Moreover, we prove the central limit theorem for the overlap in this region. The present paper provides an alternative proof of the replica-symmetric free energy formula in the de Almeida--Thouless region, which was recently established by Lopatto [arXiv:2604.11921]. An advantage of our approach is that it establishes an explicit quantitative bound and yields the replica-symmetric free energy formula as a consequence. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.
Related
- The quantum Almeida-Thouless line in the self-overlap-corrected quantum Sherrington-Kirkpatrick model
- Full replica symmetry breaking in the Sherrington-Kirkpatrick model
- Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model
- Sampling from the Sherrington-Kirkpatrick Model up to Criticality
- Free energy and fluctuations of generalized bipartite Sherrington-Kirkpatrick models and generalized Hopfield models
BibTeX
@misc{kusuoka2026quantitative,
title = {{A quantitative replica-symmetric bound for Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region}},
author = {Seiichiro Kusuoka and Shuta Nakajima},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.23413},
eprint = {2608.23413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23413},
}
RIS
TY - UNPB
TI - A quantitative replica-symmetric bound for Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region
AU - Seiichiro Kusuoka
AU - Shuta Nakajima
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.23413
UR - https://arxiv.org/abs/2608.23413
AB - We consider the Sherrington--Kirkpatrick model with inverse temperature β>0 and deterministic external field h>0. Let q be the replica-symmetric fixed point: q=𝔼tanh^2 (h+β√q Z), where Z is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida--Thouless region: β^2𝔼sech^4(h+β√q Z) <1, the overlap satisfies the concentration: 𝔼⟨ (R_12−q)^2⟩=O(N^−1). As a consequence, we obtain an O(N^−1) replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. Moreover, we prove the central limit theorem for the overlap in this region. The present paper provides an alternative proof of the replica-symmetric free energy formula in the de Almeida--Thouless region, which was recently established by Lopatto [arXiv:2604.11921]. An advantage of our approach is that it establishes an explicit quantitative bound and yields the replica-symmetric free energy formula as a consequence. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.
ER -