Replica approach to the generalized Rosenzweig-Porter model

Davide Venturelli, Leticia F. Cugliandolo, Grégory Schehr, Marco Tarzia

Published 2022-09-23Version 1

The generalized Rosenzweig--Porter model arguably constitutes the simplest random matrix ensemble displaying a non-ergodic delocalized phase, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size $N$ of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. We confirm our results with numerical tests.