Abstract
We consider the spherical spin glass model defined by a combination of the pure 2-spin spherical Sherrington–Kirkpatrick Hamiltonian and the ferromagnetic Curie–Weiss Hamiltonian. In the large system limit, there is a two-dimensional phase diagram with respect to the temperature and the coupling strength. The phase diagram is divided into three regimes; ferromagnetic, paramagnetic, and spin glass regimes. The fluctuations of the free energy are known in each regime. In this paper, we study the transition between the ferromagnetic regime and the paramagnetic regime in a critical scale.
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Notes
In [5], we consider the case when the diagonal entries of \(M\) have mean \(\frac{J'}{N}\) and the off-diagonal entries have mean \(\frac{J}{N}\) where J and \(J'\) are allowed to be different. However, in this case, \(M=\frac{1}{\sqrt{N}} + \frac{J}{N}\mathbf {1} \mathbf {1}^T + \frac{J'-J}{N} I\) where I is the identity matrix. This only shifts all eigenvalues by a deterministic small number. As we will see in Remark 2.1, it is not more general than the case with \(J' = J\).
Even though it is stated in Lemma 4.2 of [5] that the lemma holds for sufficiently small \(\delta > 0\), the proof of it is valid for any \(\delta > 0\), and we use \(\delta > 2\) for our purpose.
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The work of Jinho Baik was supported in part by NSF Grant DMS-1664692 and the Simons Fellows program. The work of Ji Oon Lee was supported in part by Samsung Science and Technology Foundation Project Number SSTF-BA1402-04.
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Baik, J., Lee, J.O. & Wu, H. Ferromagnetic to Paramagnetic Transition in Spherical Spin Glass. J Stat Phys 173, 1484–1522 (2018). https://doi.org/10.1007/s10955-018-2150-6
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DOI: https://doi.org/10.1007/s10955-018-2150-6