It has been recently shown [1] that the $N*N$ generalized Rosenzweig-Porter random matrix model [2] can be in three distinctly different phases: ergodic, fractal and localized, depending on the scaling parameter \gamma in the variance $N^{-\gamma}$ of the i.i.d. Gaussian off-diagonal matrix elements.  Several models of disordered systems can be reduced to the logarithmically-normal  Rosenzweig-Porter ensemble in which i.i.d. off-diagonal matrix elements  have a log-normal distribution with the variance proportional to $\ln N$, while the diagonal matrix elements remain Gaussian distributed with the variance 1. In this case the typical and the averaged values of off-diagonal matrix elements have different scaling with $N$. In the present work we address the question how the tailed distributions of this type alter the phases and phase transitions in the eigenvector statistics. 

[1] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas and M. Amini, A random matrix model with localization and ergodic transitions, New J. Phys. 17, 122002 (2015)
[2] N. Rosenzweig and C. E. Porter, "Repulsion of energy levels" in complex atomic spectra, Phys. Rev. 120, 1698 (1960)