Limits of characters of wreath products S n (T) of a compact group T with the symmetric groups and characters of S ∞ (T) , II -- From a viewpoint of probability theory

Abstract

This paper is the second part of our study on limiting behavior of characters of wreath products ${\mathfrak{S}}_n(T)$ of compact group $T$ as $n\to \infty$ and its connection with characters of ${\mathfrak{S}}_{\infty }(T)$ . Contrasted with the first part, which has a representation-theoretical flavor, the approach of this paper is based on probabilistic (or ergodic-theoretical) methods. We apply boundary theory for a fairly general branching graph of infinite valencies to wreath products of an arbitrary compact group $T$ . We show that any character of ${\mathfrak{S}}_{\infty }(T)$ is captured as a limit of normalized irreducible characters of ${\mathfrak{S}}_n(T)$ as $n\to \infty$ along a path on the branching graph of ${\mathfrak{S}}_{\infty }(T)$ . This yields reconstruction of an explicit character formula for ${\mathfrak{S}}_{\infty }(T)$ .