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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.
Physical Review. E
|September 18, 2020
Summary
This study introduces a novel exponentiation formula connecting classical extreme value laws (Fréchet, Gumbel, Weibull) with their free counterparts. It establishes a new random matrix theory framework for analyzing extreme matrices and their properties.
Area of Science:
- Probability Theory
- Random Matrix Theory
- Extreme Value Theory
Background:
- Classical extreme value theory classifies extreme events using Fréchet, Gumbel, and Weibull laws (Fisher-Tippet-Gnedenko).
- Free probability theory, developed by Ben Arous and Voiculescu, introduced analogous free extreme laws.
Purpose of the Study:
- To establish an explicit link between classical and free extreme value laws.
- To develop a random matrix formalism for studying extreme matrices.
- To demonstrate the practical application of free extreme laws in random matrix ensembles.
Main Methods:
- Development of a novel exponentiation formula.
- Construction of an extreme random matrix formalism.
- Explicit calculations on random matrix ensembles.
- Derivation of an exact mapping between free extreme laws and the Peak-over-Threshold method.
Main Results:
- An explicit exponentiation formula linking classical extreme laws (Fréchet, Gumbel, Weibull) and free extreme laws (free Fréchet, free Gumbel, free Weibull) was derived.
- A new random matrix formalism for extreme matrices was developed, enabling refined analyses.
- Examples of all three free extreme laws were demonstrated in various random matrix ensembles.
- An exact mapping demonstrating the equivalence of free extreme laws to the Peak-over-Threshold method was presented.
Conclusions:
- The study provides a unified framework for understanding extreme value distributions in both classical and free probability.
- The developed random matrix formalism offers new tools for analyzing extreme matrix properties.
- The equivalence to the Peak-over-Threshold method highlights the practical relevance of free extreme laws.
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