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We study random covariance matrices whose entries have exploding moments meaning that the ratio between their k-th moment and the k-th power of their standard deviation goes to infinity with the size of the matrix. We compute an asymptotic expansion of the limiting spectral measure in the critical regime when this measure is close to the Marchenko-Pastur distribution. Explicit computations are given in the two classical cases of Bernoulli and truncated heavy tailed entries. MSC 2010 Classification: 05C80; 60B20.

Spectral asymptotic expansion of Wishart matrices with exploding moments