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We consider Hermitian matrices with i.i.d. entries whose tail probabilities behave like for some and . We establish a large deviation principle for the empirical spectral measure of with speed with a good rate function that is finite only if is of the form for some probability measure on , where denotes the free convolution and is Wigner’s semicircle law. We obtain explicit expressions for in terms of the th moment of . The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.