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

In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble $GCy$, whose eigenvalues PDF is given by $$\text{const}\cdot \prod _{1\le j<k\le N}(x_j-x_k{)}^2\prod _{j=1}^N(1+i{x}_j{)}^{-s-N}(1-i{x}_j{)}^{-\bar{s}-N}d{x}_j,$$where $s$ is a complex number such that $\mathrm{\Re }(s)>-1\mathrm{/}2$ and where $N$ is the size of the matrix ensemble. Using results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that for this ensemble, the largest eigenvalue divided by $N$ converges in law to some probability distribution for all $s$ such that $\mathrm{\Re }(s)>-1\mathrm{/}2$. Using results by Forrester and Witte \cite{Forrester-Witte2} on the distribution of the largest eigenvalue for fixed $N$, we also express the limiting probability distribution in terms of some non-linear second order differential equation. Eventually, we show that the convergence of the probability distribution function of the re-scaled largest eigenvalue to the limiting one is at least of order $(1\mathrm{/}N)$.