A permutation model for free random variables and its classical analogue

TL;DR

This paper introduces a generalized permutation model for free and classical random variables, providing new discrete approximations of Wiener chaos and explicit asymptotic independence examples.

Contribution

It extends Biane's permutation model to a classical analogue using subsets with symmetric difference, offering novel discrete approximations and combinatorial insights.

Findings

Explicit asymptotic free and independent matrices constructed

Moments and cumulants expressed via noncrossing pairings

New combinatorial applications demonstrated

Abstract

In this paper, we generalize a permutation model for free random variables which was first proposed by Biane in \cite{biane}. We also construct its classical probability analogue, by replacing the group of permutations with the group of subsets of a finite set endowed with the symmetric difference operation. These constructions provide new discrete approximations of the respective free and classical Wiener chaos. As a consequence, we obtain explicit examples of non random matrices which are asymptotically free or independent. The moments and the free (resp. classical) cumulants of the limiting distributions are expressed in terms of a special subset of (noncrossing) pairings. At the end of the paper we present some combinatorial applications of our results.