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Extreme matrices or how an exponential map links classical and free extreme laws
1Institute of Theoretical Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland.
Abstract:
Using our proposed approach to describe extreme matrices, we find an explicit exponentiation formula linking the classical extreme laws of Fréchet, Gumbel, and Weibull given by the Fisher-Tippet-Gnedenko classification and free extreme laws of free Fréchet, free Gumbel, and free Weibull of Ben Arous and Voiculescu. 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.
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