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Using the proposed by us thinning approach to describe extreme matrices, we find an explicit exponentiation formula linking classical extreme laws of Frechet, Gumbel and Weibull given by Fisher-Tippet-Gnedenko classification and free extreme laws of free Frechet, free Gumbel and free Weibull by Ben Arous and Voiculescu [1]. We also develop an extreme random matrix formalism, in which refined questions about extreme matrices can be answered. In particular, we demonstrate explicit calculations for several more or less known random matrix ensembles, providing examples of all three free extreme laws. Finally, we present an exact mapping, showing the equivalence of free extreme laws to the Peak-Over-Threshold method in classical probability.
We consider random stochastic matrices $M$ with elements given by $M_{ij}=|U_{ij}|^2$, with $U$ being uniformly distributed on one of the classical compact Lie groups or associated symmetric spaces. We observe numerically that, for large dimensions,
How to calculate the exponential of matrices in an explicit manner is one of fundamental problems in almost all subjects in Science. Especially in Mathematical Physics or Quantum Optics many problems are reduced to this calculation by making use of
Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of the form $P(Yleq y mid X >t) $ is challenging when $t$ is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the ran
This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erd{H o}s-Schlein-Yau d
We study the joint probability density of the eigenvalues of a product of rectangular real, complex or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restricti