Localization Transitions in Effective Random Matrix Models

• Applicant: Dr. Per von Soosten
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 407999979

• Final Report Year: 2021

Final Report Abstract

In this project we investigated the use of dynamical methods to derive fixed-point equations for some mean-field models in statistical physics. Our original aim was to establish that the amplitudes of typical eigenvectors of certain effective random matrix models are spread evenly throughout the entire volume. We succeeded in proving this for the ultrametric ensemble, but a proof for more general models continues to be an open problem. We showed that closely related methods are also applicable to Wigner matrices, where we gave a new proof of the weak local semicircle law, and to mean-field ferromagnetic spin systems, where we derived a general form of the classical mean-field equations. Finally, we conducted a dynamical study of the Thouless-Anderson-Palmer equations for the Sherrington-Kirkpatrick spin glass and derived their analogues for higher-order correlation functions at high temperature.

Publications

• Random characteristics for Wigner matrices. Electron. Commun. Probab., 24(75), 1-12 (2019)
  von Soosten, Per & Warzel, Simone
  
• Dynamical approach to the TAP equations for the Sherrington-Kirkpatrick model. J. Stat. Phys., 183, 35 (2021). C. Brennecke and P. von Soosten. On the mean-field equations for ferromagnetic spin systems. Lett. Math. Phys., 111, 108 (2021)
  Adhikari, Arka; Brennecke, Christian; von Soosten, Per & Yau, Horng-Tzer