Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model with Ferromagnetic Interaction

2016/07/19 by Jinho Baik, Ji Oon Lee · 43 citations
Mathematics · #Combinatorics #Condensed matter physics #Coupling constant #Eigenvalues and eigenvectors #Ferromagnetism #Hamiltonian (control theory) #Limit (mathematics) #Limiting #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematical physics #Mathematics #Phase transition #Physics #Quantum mechanics #Random Matrices and Applications #Random matrix #Rank (graph theory) #Spherical model #Spin glass #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.1007/s00023-017-0562-5

published in Annales Henri Poincaré 18(6), 1867-1917 (Birkhäuser) · 45 pages, references added

arxiv created 2016/07/19 · openalex publication_date 2017/02/11 · arxiv updated 2018/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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.

Citations

• The Thermodynamic Limit in Mean Field Spin Glass Models
• The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case
• Topology Trivialization and Large Deviations for the Minimum in the Simplest Random Optimization
• On the overlap in the multiple spherical SK models
• On fluctuations of eigenvalues of random Hermitian matrices
• Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge
• The extremal process of critical points of the pure p-spin spherical\n spin glass model