Higher Order Moments and Free Cumulants of Complex Wigner Matrices

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Authors
Munoz George, Daniel
Keyword
Wigner , Random matrix theory , Free Probability , Higher order moments , Free cumulants , Cumulants
Abstract
The main object of this thesis is a random matrix model known as the Wigner model. We investigate the higher order moments and free cumulants of a complex Wigner matrix. It is well known that the first order moments and free cumulants of this model are described by the Catalan numbers and the semicircle law, this constitutes part of the work done by Wigner in the 1950s. Later on, in 2020, Male, Mingo, Péché and Speicher answered this question for the second order case, these are known as the fluctuation moments. In this thesis, we provide a formula for the third order moments in terms of the set of partitioned permutations. Furthermore, we also present a simple expression for the third order free cumulants. In Chapter 6 we talk about the progress that has been made for any higher order case. We prove that the higher order moments and free cumulants exist as long as we ask for suitable conditions.
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