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In this paper, we are interested in the asymptotic properties for the largesteigenvalue of the Hermitian random matrix ensemble, called the GeneralizedCauchy ensemble $GCy$, whose eigenvalues PDF is given by$$\textrm{const}\cdot\prod_{1\leq j<k\leq N}(x_j-x_k)^2\prod_{j=1}^N(1+ix_j)^{-s-N}(1-ix_j)^{-\bar{s}-N}dx_j,$$where $s$ is a complex number suchthat $\Re(s)>-1/2$ and where $N$ is the size of the matrix ensemble. Usingresults by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove thatfor this ensemble, the largest eigenvalue divided by $N$ converges in law tosome probability distribution for all $s$ such that $\Re(s)>-1/2$. Usingresults by Forrester and Witte \cite{Forrester-Witte2} on the distribution ofthe largest eigenvalue for fixed $N$, we also express the limiting probabilitydistribution in terms of some non-linear second order differential equation.Eventually, we show that the convergence of the probability distributionfunction of the re-scaled largest eigenvalue to the limiting one is at least oforder $(1/N)$.

Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble