Lokalisierungsübergänge in Effektiven Zufallsmatrixmodellen
Zusammenfassung der Projektergebnisse
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.
Projektbezogene Publikationen (Auswahl)
- Random characteristics for Wigner matrices. Electron. Commun. Probab., 24(75), 1-12 (2019)
P. von Soosten and S. Warzel
(Siehe online unter https://doi.org/10.1214/19-ECP278) - 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)
A. Adhikari, C. Brennecke, P. von Soosten and H.-T. Yau
(Siehe online unter https://doi.org/10.1007/s10955-021-02773-7)
