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This work provide a thorough study of L\'evy or heavy-tailed random matrices(LM). By analysing the self-consistent equation on the probability distributionof the diagonal elements of the resolvent we establish the equation determiningthe localisation transition and obtain the phase diagram of LMs. Usingarguments based on super-symmetric field theory and Dyson Brownian motion weshow that the eigenvalue statistics is the same one of the Gaussian OrthogonalEnsemble in the whole delocalised phase and is Poisson in the localised phase.Our numerics confirms these findings, valid in the limit of infinitely largeLMs, but also reveals that the characteristic scale governing finite sizeeffects diverges much faster than a power law approaching the transition and isalready very large far from it. This leads to a very wide cross-over region inwhich the system looks as if it were in a mixed phase. Our results, togetherwith the ones obtained previously, provide now a complete theory of L\'evymatrices.

Level statistics and localization transitions of L\'evy matrices