A phase transition for tails of the free multiplicative convolution powers

Abstract

We study the behavior of the tail of a measure ${\mu }^{⊠t}$, where $⊠t$ is the $t$-fold free multiplicative convolution power for $t\ge 1$. We focus on the case where $\mu$ is a probability measure on the positive half-line with a regularly varying tail i.e. of the form $x^{-\alpha }L(x)$, where $L$ is slowly varying. We obtain a phase transition in the behavior of the tail of ${\mu }^{⊞t}$ between regimes $\alpha <1$ and $\alpha >1$. Our main tool is a description of the regularly varying tails of $\mu$ in terms of the behavior of the corresponding $S$-transform at $0^{-}$. We also describe the tails of $⊠$ infinitely divisible measures in terms of the tails of corresponding Lévy measure, treat symmetric measures with regularly varying tails and prove the free analog of the Breiman lemma.

arXiv comment44 pages