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The space of one-dimensional disordered interacting quantum models displayinga Many-Body-Localization Transition seems sufficiently rich to produce criticalpoints with level statistics interpolating continuously between the Poissonstatistics of the Localized phase and the Wigner-Dyson statistics of theDelocalized Phase. In this paper, we consider the strong disorder limit of theMBL transition, where the critical level statistics is close to the Poissonstatistics. We analyse a one-dimensional quantum spin model, in order todetermine the statistical properties of the rare extensive resonances that areneeded to destabilize the MBL phase. At criticality, we find that theentanglement entropy can grow with an exponent $0<\alpha < 1$ anywhere betweenthe area law $\alpha=0$ and the volume law $\alpha=1$, as a function of theresonances properties, while the entanglement spectrum follows the strongmultifractality statistics. In the MBL phase near criticality, we obtain thesimple value $\nu=1$ for the correlation length exponent. Independently of thestrong disorder limit, we explain why for the Many-Body-Localization transitionconcerning individual eigenstates, the correlation length exponent $\nu$ is notconstrained by the usual Harris inequality $\nu \geq 2/d$, so that there is notheoretical inconsistency with the best numerical measure $\nu = 0.8 (3)$obtained by D. J. Luitz, N. Laflorencie and F. Alet, Phys. Rev. B 91, 081103(2015).

Many Body Localization Transition in the strong disorder limit : entanglement entropy from the statistics of rare extensive resonances