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arXiv · 2604.07130

Existence of a Phase Transition in the One-Dimensional Ising Spin Glass Model with Long-Range Interactions on the Nishimori Line

Manaka Okuyama·Masayuki Ohzeki
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Abstract

Dyson [Commun. Math. Phys. 12, 91 (1969)] rigorously proved the existence of a phase transition in the one-dimensional Ising model with long-range interactions of the form $r^{-\alpha}$ for $1 < \alpha < 2$. In the present study, we extend this result to the Ising spin glass model with Gaussian disorder on the Nishimori line. Following Dyson's method, we first prove the existence of long-range order at finite low temperatures in the Dyson hierarchical Ising spin glass model on the Nishimori line, with power-law-like interactions $J(r) \sim r^{-\alpha}$ for $1 < \alpha < 3/2$. The key ingredients of the proof are the interpolation method developed in the rigorous analysis of mean-field spin glass models, the Gibbs--Bogoliubov inequality on the Nishimori line, and the Tsirelson--Ibragimov--Sudakov inequality (Gaussian concentration inequality). We then use the Griffiths inequality on the Nishimori line to rigorously establish the existence of a phase transition in the one-dimensional Ising spin glass model with long-range interactions on the Nishimori line for $1 < \alpha < 3/2$. For $3/2 \le \alpha \le 2 $, the existence of a phase transition remains an open problem.

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BibTeXRIS

Manaka Okuyama, Masayuki Ohzeki. 2026-04-08. Existence of a Phase Transition in the One-Dimensional Ising Spin Glass Model with Long-Range Interactions on the Nishimori Line. https://doi.org/10.1088/1751-8121%2Fae8c95

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