Large deviations of the top eigenvalue of large Cauchy random matrices

Satya N. Majumdar 1, Gregory Schehr 1, Dario Villamaina 1, Pierpaolo Vivo 1

Journal of Physics A: Mathematical and Theoretical 46 (2013) 022001

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 ($\beta =1$) and symplectic ($\beta=4$) 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.

  • 1 : Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS)
    CNRS : UMR8626 – Université Paris XI - Paris Sud