Limits of characters of wreath products \mathfrak{S}_n(T) of a compact group T with the symmetric groups and characters of \mathfrak{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 → ∞ and its connection with characters of \mathfrak{S}_∞(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}_∞(T) is captured as a limit of normalized irreducible characters of \mathfrak{S}_n(T) as n → ∞ along a path on the branching graph of \mathfrak{S}_∞(T). This yields reconstruction of an explicit character formula for \mathfrak{S}_∞(T).