The Distribution of Polynomials in Monotone Independent Elements

Abstract

Building on the work of Arizmendi and Celestino (2021), we derive the $\ast$-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers $\hat{I}\pm$ and ${\hat{I}}^2$, we derive explicitly the $\ast$-distribution of $p_{\hat{I}\pm ,{\hat{I}}^2}=\hat{I}\pm ab+{\hat{I}}^{2}ba$ whenever $a$ and $b$ are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

18 pages