Fluctuation of the Free Energy of Sherrington-Kirkpatrick Model with Curie-Weiss Interaction: The Paramagnetic Regime

摘要

We consider a spin system containing pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by Baik and Lee (Ann Henri Poincare 18(6):1867-1917, 2017) and Baik et al. (J Stat Phys 173(5):1484-1522, 2018) which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to Baik and Lee (Ann Henri Poincare 18(6):1867-1917, 2017) in the "paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by Banerjee (J Probab 23:28, 2018). The proof of the approximation by the linear spectral statistics part is close to Banerjee and Ma (arXiv:1705.05305, 2017).