Since the pioneering works of Wishart and Wigner on random matrices, matrices with independent entries with finite moments have been intensively studied.Not only it was shown that their spectral measure converges to the semicircle law, but fluctuations both global and local were analyzed in fine details.More recently, the domain of universality of these results was investigated, in particular by Erdos-Yau et al and Tao-Vu et al.This survey article takes the opposite point of view by considering matrices which are not in the domain of universality of Wigner matrices: they have independent entries but with heavy tails.We discuss the properties of these matrices.They are very different from Wigner matrices: the limit law of the spectral measure is not the semi-circle distribution anymore, the global fluctuations are stronger and the local fluctuations may undergo a transition and remain rather mysterious.
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