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A permutation model for free random variables and its classical analogue

In this paper, we generalize a permutation model for free random variables which was first proposed by 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 nonrandom matrices which are asymptotically free or independent. The moments and the free (respectively 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.