  The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries
arguably constitutes the simplest random matrix ensemble displaying a phase
with fractal eigenstates, 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. Interestingly, the
same scaling function is found to describe the level compressibility of the
complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results
with numerical tests.
