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Abstract: 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-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 $\Res>-1-2$. 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 $1-N$.



Autor: Joseph Najnudel, Ashkan Nikeghbali, Felix Rubin

Fuente: https://arxiv.org/







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