具有铁磁相互作用的Sherrington-Kirkpatrick模型的自由能涨落
Fluctuations of the free energy of the Sherrington-Kirkpatrick model with ferromagnetic interaction
AI总结：
该研究针对带铁磁相互作用的Sherrington-Kirkpatrick模型，推导其铁磁区自由能的中心极限定理，完善纯2自旋模型的相图相关涨落结论，是球形SKFI模型涨落结果的伊辛类比。
AI中文摘要：
我们研究带有额外Curie-Weiss铁磁相互作用的Sherrington-Kirkpatrick模型（SKFI），其在无序强度β和铁磁耦合γ平面内的相图包含顺磁区、铁磁区和自旋玻璃区。我们的主要结果是铁磁区自由能的中心极限定理：在N^{-1/2}尺度下，涨落与由极限磁化强度确定的有效外场中Sherrington-Kirkpatrick模型的涨落一致。我们利用Chen、Dey和Panchenko的涨落理论，对一般混合偶p自旋相互作用证明了该结论。对于纯2自旋模型，我们完善了相图的图景：对Banerjee此前获得的顺磁区阶为N^{-1}的高斯涨落给出新证明，该证明在临界窗口内有效；确定了分隔顺磁区与铁磁区的临界线γ=1+θN^{-1/2}上的极限分布；并证明在自旋玻璃区，涨落与零场SK模型的涨落一致，前提是满足广泛认可的方差发散假设。我们的结果是Baik和Lee关于球形SKFI模型涨落结果的伊辛类比。
英文摘要：
We study the Sherrington-Kirkpatrick model with an additional Curie-Weiss ferromagnetic interaction (SKFI), whose phase diagram in the plane of the disorder strength $β$ and the ferromagnetic coupling $γ$ consists of a paramagnetic, a ferromagnetic, and a spin glass region. Our main result is a central limit theorem for the free energy in the ferromagnetic regime: at scale $N^{-1/2}$, the fluctuations coincide with those of the Sherrington-Kirkpatrick model in an effective external field determined by the limiting magnetization. We prove this for general mixed even $p$-spin interactions, using the fluctuation theory of Chen, Dey, and Panchenko. For the pure 2-spin model we complete the picture of the phase diagram: we give a new proof of the order-$N^{-1}$ Gaussian fluctuations in the paramagnetic regime, obtained earlier by Banerjee, valid up to the critical window; we determine the limiting distribution on the critical line $γ= 1 + θN^{-1/2}$ separating the paramagnetic and ferromagnetic regimes; and we show that in the spin glass regime the fluctuations coincide with those of the zero-field SK model under a widely believed variance-divergence assumption. Our results form the Ising analogue of the fluctuation results of Baik and Lee for the spherical SKFI model.
