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Large deviations of the top eigenvalue of large Cauchy random matrices

We compute analytically the probability density function (pdf) of the largest eigenvalue $λ_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($β= 2$), orthogonal ($β=1$) and symplectic ($β=4$) ensembles of such heavy-tailed random matrices. We show that a central non-Gaussian regime for $λ_{\max} \sim \mathcal{O}(N)$ is flanked by large deviation tails on both sides which we compute here exactly for any value of $β$. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors of the pdf in the central regime, which generalizes the Tracy-Widom distribution known for Gaussian ensembles, both at small and large arguments and for any $β$. Our analytical results are confirmed by numerical simulations.