标题： 球形Sherrington-Kirkpatrick模型自由能的涨落与铁磁相互作用 显示英文标题

标题： Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with ferromagnetic interaction

摘要： 我们考虑一个具有纯2自旋球形的Sherrington-Kirkpatrick哈密顿量与铁磁Curie-Weiss相互作用的球形自旋系统。该系统在温度和耦合常数方面表现出二维相变。我们计算了所有远离临界值参数的自由能的极限分布。零温度情况对应于一个秩1尖峰随机对称矩阵的最大特征值的著名相变。作为中间步骤，我们建立了秩1尖峰随机对称矩阵的线性统计量的中心极限定理。 显示英文摘要

摘要： We consider a spherical spin system with pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank 1 spiked random symmetric matrices.