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Description

Take a square $N\times N$ symmetric matrix $X_N$ with real i.i.d random entries above the diagonal with law $\mu$ such that $\mu(x2)=1$. Then, if $\lambda_i, 1\le i\le N$ denote the eigenvalues of $X_N$, Wigner's theorem asserts that the spectral measure $N^{-1}\sum_{i=1}^N \delta_{\lambda_i/sqrt{N}}$ converges weakly in expectation and almost surely towards the semi-circle law. We study what happens when $\mu$ has no finite second moment, but belong to the domain of attraction of an $\alpha$-stable law. In particular, we show the weak convergence of $\E[N^{-1}\sum_{i=1}^N \delta_{\lambda_i /N^{1/\alpha}}]$ towards a symmetric law with heavy tail.