Abstract In this paper, we study a mean-field spin model with three- and two-body interactions. In a recent paper (Ann Henri Poincaré, 2024) by Contucci, Mingione and Osabutey, the equilibrium measure for large volumes was shown to have three pure states, two with opposite magnetization and an unpolarized one with zero magnetization, merging at the critical point. The authors proved a central limit theorem for the suitably rescaled magnetization. The aim of our paper is presenting a prove of a central limit theorem for the rescaled magnetization applying the exchangeable pair approach due to Stein. Moreover we prove (non-uniform) Berry–Esseen bounds, a concentration inequality, Cramér-type moderate deviations and a moderate deviations principle for the suitably rescaled magnetization. Interestingly we analyze Berry–Esseen bounds in case the model-parameters $$(K_n,J_n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> converge to the critical point (0, 1) on lines with different slopes and with a certain speed, and obtain new limiting distributions and thresholds for the speed of convergence.

Peter Eichelsbacher

Journal of Statistical Physics

《统计物理学杂志》在统计物理学的所有领域以及与物理系统中的集体现象有关的相关领域发表原创和特邀评论论文。

The Journal of Statistical Physics publishes original research articles and invited reviews in all areas of statistical physics, with a particular emphasis on topics related to collective phenomena in physical systems.

《Journal of Statistical Physics》致力于发表统计物理学各个领域的原创研究论文和特邀评论，特别关注与物理系统中的集体现象相关的研究。

Category	Quartile
物理与天体物理	3
物理与天体物理, 物理数学物理	3

Stein’s Method and a Cubic Mean-Field Model

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Category	Quartile
PHYSICS, MATHEMATICAL	3