# Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble - Mathematics > Probability

Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble - Mathematics > Probability - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

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/