Nonlinear Random Matrix Statistics, symmetric functions and hyperdeterminants

TL;DR

This paper develops formulas for nonlinear eigenvalue statistics of invariant random matrices across Dyson classes, utilizing hyperdeterminants and symmetric functions, with applications to quantum transport in chaotic systems.

Contribution

It introduces new hyperdeterminant-based formulas and efficient symmetric function methods for nonlinear eigenvalue statistics in random matrix theory.

Findings

Derived general hyperdeterminant formulas for Dyson class β=2.

Provided computationally efficient symmetric function expansions.

Extended quantum transport results to broader random matrix models.

Abstract

Nonlinear statistics (i.e. statistics of permanents) on the eigenvalues of invariant random matrix models are considered for the three Dyson's symmetry classes $\beta=1,2,4$. General formulas in terms of hyperdeterminants are found for $\beta=2$. For specific cases and all $\beta$s, more computationally efficient results are obtained, based on symmetric functions expansions. As an application, we consider the case of quantum transport in chaotic cavities extending results from [D.V. Savin, H.-J. Sommers and W. Wieczorek, {\it Phys. Rev. B} {\bf 77}, 125332 (2008)].