Large deviations of the top eigenvalue of large Cauchy random matrices

We compute analytically the probability density function (pdf) of the largest eigenvalue ${\lambda }_{\max }$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($\beta =2$), orthogonal ($\beta =1$)andsymplectic($\beta=4$) ensembles of such heavy-tailed random matrices. We show that a central non-Gaussian regime for $\lambda_{\max} \sim $\mathcal{O}$(N)$ is flanked by large deviation tails on both sides which we compute here exactly for any value of $\beta$. 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 $\beta$. Our analytical results are confirmed by numerical simulations.