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Gaussian Rosenzweig-Porter (GRP) random matrix ensemble is the only one inwhich the robust multifractal phase and ergodic transition have a status of amathematical theorem. Yet, this phase in GRP model is oversimplified: thespectrum of fractal dimensions is degenerate and the mini-band in the localspectrum is not multifractal. In this paper we suggest an extension of the GRPmodel by adopting a logarithmically-normal (LN) distribution of off-diagonalmatrix elements. A family of such LN-RP models is parametrized by a symmetryparameter $p$ and it interpolates between the GRP at $p\rightarrow 0$ and Levyensembles at $p\rightarrow\infty$. A special point $p=1$ is shown to be thesimplest approximation to the Anderson localization model on a random regulargraph.We study in detail the phase diagram of LN-RP model and show that $p=1$is a tricritical point where the multifractal phase first collapses. Thiscollapse is shown to be unstable with respect to the truncation of thelog-normal distribution. We suggest a new criteria of stability of thenon-ergodic phases and prove that the Anderson transition in LN-RP model isdiscontinuous at all $p>0$.

Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble