We compute analytically the large deviation tails of the probability density function (pdf) of the top eigenvalue λmax in rotationally invariant and heavy-tailed Cauchy ensembles of N × N matrices for any Dyson index β > 0, where β = 1, 2, 4 correspond, respectively, to orthogonal, unitary and symplectic ensembles. Furthermore, we show that these large deviation tails flank a central non-Gaussian regime for on both sides. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors for any β of the pdf in the central regime, which generalizes the Tracy-Widom distribution known for Gaussian ensembles. Our analytical results are confirmed by numerical simulations.
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