Permutation model for semi-circular systems and quantum random walks
Let \(E\) be a countable set with a fixed element \(e_0\), \(S(E)\) the group of permutations with finite support of \(E\) and \((e^j_n)_{n \in \mathbb{N}}\) for \(j \in J\) be disjoint sequences in \(E\). Let \(\tau^j_n\) be the transposition \((e_0, e^j_n)\), it is proved that the family of noncommutative random variables \((a^j_N = {1 \over \sqrt N} \sum^N_{n = 1} \tau^j_n\), \(j \in J) \) in the group-von Neumann algebra of \(S(E)\) converges in distribution towards a semicircular system. By using some results on random walks on the dual of unitary groups a relation between this result and the random matrix model for semicircular systems of Voiculescu is exhibited.
scientific article; zbMATH DE number 912150