Scaled limit for the largest eigenvalue from the generalized Cauchy random matrix ensemble

In this talk, 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 $\mathrm{\Re }(s)>-1/2$ and where $N$ is the size of the matrix ensemble. We will see that for this ensemble, the appropriately rescaled largest eigenvalue converges in law.

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 $(1/N)$.